#set document(title: "7.2 Graphing Polynomial Functions", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 7.2#h(0.6em)Graphing Polynomial Functions In Polynomial Functions, we considered applications of polynomial functions. Although most applications use only a portion of the graph of a particular polynomial, we can learn a lot about these functions by taking a more global view of their behavior. === Classifying Polynomials by Degree The graph of a polynomial function depends first of all on its degree. We have already studied the graphs of polynomials of degrees #math.equation(block: false, alt: "0")[$0$], #math.equation(block: false, alt: "1")[$1$], and #math.equation(block: false, alt: "2")[$2$]. - A polynomial of degree #math.equation(block: false, alt: "0")[$0$] is a constant, and its graph is a horizontal line. An example of such a polynomial function is #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 3")[$f ( x ) = 3$]. - A polynomial of degree #math.equation(block: false, alt: "1")[$1$] is a linear function, and its graph is a straight line. The function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 x minus 3")[$f ( x ) = 2 x − 3$] is an example of a polynomial of degree #math.equation(block: false, alt: "1")[$1$].#figure(figph[horizontal line and line of nonzero slope], alt: "horizontal line and line of nonzero slope", caption: none) - Quadratic functions, such as #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals minus 2 x squared plus 6 x plus 8")[$f ( x ) = − 2 x^(2) + 6 x + 8$], are polynomials of degree #math.equation(block: false, alt: "2")[$2$]. The graph of every quadratic function is a parabola, with the same basic shape as the standard parabola, #math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$]. It has one turning point, where the graph changes from increasing to decreasing or vice versa. The turning point of a parabola is the same as its vertex.#figure(figph[basic parabola and parabola opening down], alt: "basic parabola and parabola opening down", caption: none) === Cubic Polynomials Do the graphs of all #strong[cubic], or third-degree, polynomials have a basic shape in common? We can graph a few examples and find out. Unlike the basic parabola, the graph of #math.equation(block: false, alt: "y equals x cubed")[$y = x^(3)$] is always increasing. At the origin, however, it changes from concave down to concave up. A point where the graph changes concavity is called an #strong[inflection point]. #examplebox("Example 1")[][ Graph the cubic polynomial #math.equation(block: false, alt: "P open parenthesis x close parenthesis equals x cubed minus 4 x")[$P ( x ) = x^(3) − 4 x$] and compare its graph with that of the basic cubic, #math.equation(block: false, alt: "y equals x cubed")[$y = x^(3)$]. #solutionbox[ The graph of the basic cubic is shown in figure (a) below. To help us understand the graph of the polynomial #math.equation(block: false, alt: "P open parenthesis x close parenthesis equals x cubed minus 4 x")[$P ( x ) = x^(3) − 4 x$], we will evaluate the function to make a table of values. We can do this by hand or use the Table feature on the graphing calculator. #figure(figph[graphs of basic cubic and cubic with two turning points], alt: "graphs of basic cubic and cubic with two turning points", caption: none) #figure(table( columns: 8, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "3")[$3$]]), [#math.equation(block: false, alt: "P open parenthesis x close parenthesis")[$P ( x )$]], [#math.equation(block: false, alt: "minus 15")[$− 15$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "15")[$15$]], )) The graph of #math.equation(block: false, alt: "P open parenthesis x close parenthesis equals x cubed minus 4 x")[$P ( x ) = x^(3) − 4 x$] is shown in figure (b). It is not exactly the same shape as the basic cubic—it has two turning points—but it is similar, especially at the edges of the graphs. ] ] At an inflection point, a graph changes \_\_\_\_\_ #solutionbox[ A graph changes concavity at an inflection point. ] At an inflection point, a graph changes + direction. + concavity. + sign. + all of these. Despite the differences in the central portions of the two graphs, they exhibit similar #emph[long-term] behavior. - For very large and very small values of #math.equation(block: false, alt: "x")[$x$], both graphs look like the power function #math.equation(block: false, alt: "y equals x cubed")[$y = x^(3)$]. - The #math.equation(block: false, alt: "y")[$y$]-values increase from #math.equation(block: false, alt: "minus ∞")[$− upright(∞)$] toward zero in the third quadrant, and they increase from zero toward #math.equation(block: false, alt: "plus ∞")[$+ upright(∞)$] in the first quadrant. Or we might say that the graphs start at the lower left and extend to the upper right. All cubic polynomials display this behavior when their lead coefficients (the coefficient of the #math.equation(block: false, alt: "x cubed")[$x^(3)$] term) are positive. - Both of the graphs in Example are smooth curves without any breaks or holes. This smoothness is a feature of the graphs of all polynomial functions. - The domain of any polynomial function is the entire set of real numbers. + Complete the table of values below for #math.equation(block: false, alt: "C open parenthesis x close parenthesis equals minus x cubed minus 2 x squared plus 4 x plus 4")[$C ( x ) = − x^(3) − 2 x^(2) + 4 x + 4$].#figure(table( columns: 6, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]]), [#math.equation(block: false, alt: "y")[$y$]], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], )) #figure(table( columns: 5, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "4")[$4$]]), [#math.equation(block: false, alt: "y")[$y$]], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], )) + Graph #math.equation(block: false, alt: "y equals C open parenthesis x close parenthesis")[$y = C ( x )$] in the standard window. Compare the graph to the graphs in the Example above: What similarities do you notice? What differences?- Both non-basic cubic graphs have three #math.equation(block: false, alt: "x")[$x$]-intercepts and two turning points. \_\_\_\_\_ - Both non-basic cubic graphs have the same long-term behavior as #math.equation(block: false, alt: "y equals x cubed")[$y = x^(3)$]. \_\_\_\_\_ - The non-basic cubic function from the example has three \\(x\\)-intercepts and two turning points, but this cubic function has only one turning point. \_\_\_\_\_ - The non-basic cubic function from the example has long-term behavior like #math.equation(block: false, alt: "y equals x cubed")[$y = x^(3)$], but his cubic function has long-term behavior like #math.equation(block: false, alt: "y equals minus x cubed")[$y = − x^(3)$]. \_\_\_\_\_ #solutionbox[ + #figure(table( columns: 10, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "4")[$4$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 29")[$− 29$]], [#math.equation(block: false, alt: "minus 76")[$− 76$]], )) + A graph is below. #linebreak() Both graphs have three #math.equation(block: false, alt: "x")[$x$]-intercepts, but the function in the Example above has long-term behavior like #math.equation(block: false, alt: "y equals x cubed")[$y = x^(3)$], and this function has long-term behavior like #math.equation(block: false, alt: "y equals minus x cubed")[$y = − x^(3)$]. ] A graph for part (b): #figure(figph[cubic], alt: "cubic", caption: none) + Complete the table of values below for #math.equation(block: false, alt: "C open parenthesis x close parenthesis equals minus x cubed minus 2 x squared plus 4 x plus 4")[$" " C ( x ) = − x^(3) − 2 x^(2) + 4 x + 4$].#figure(table( columns: 10, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "4")[$4$]]), [#math.equation(block: false, alt: "y")[$y$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], )) + Graph #math.equation(block: false, alt: "y equals C open parenthesis x close parenthesis")[$" " y = C ( x )$] in the standard window. Compare the graph to the graphs in the Example above: What similarities do you notice? What differences? #solutionbox[ + #figure(table( columns: 10, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "4")[$4$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 29")[$− 29$]], [#math.equation(block: false, alt: "minus 76")[$− 76$]], )) + #figure(figph[cubic], alt: "cubic", caption: none) #linebreak() Both graphs have three #math.equation(block: false, alt: "x")[$x$]-intercepts, but the function in the Example above has long-term behavior like #math.equation(block: false, alt: "y equals x cubed")[$y = x^(3)$], and this function has long-term behavior like #math.equation(block: false, alt: "y equals minus x cubed")[$y = − x^(3)$]. ] === Quartic Polynomials Now let’s compare the long-term behavior of two #strong[quartic], or fourth-degree, polynomials. #examplebox("Example 2")[][ Graph the polynomials #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x to the power 4 minus 10 x squared plus 9")[$" " f ( x ) = x^(4) − 10 x^(2) + 9 " "$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x to the power 4 plus 2 x cubed")[$" " g ( x ) = x^(4) + 2 x^(3)$], and compare. #solutionbox[ For each function we make a table of values. #figure(table( columns: 10, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "4")[$4$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [#math.equation(block: false, alt: "105")[$105$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "minus 15")[$− 15$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "minus 15")[$− 15$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "105")[$105$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]], [#math.equation(block: false, alt: "128")[$128$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "32")[$32$]], [#math.equation(block: false, alt: "135")[$135$]], [#math.equation(block: false, alt: "384")[$384$]], )) The graphs are shown below. All the essential features of the graphs are shown in these viewing windows. The graphs continue forever in the directions indicated, without any additional twists or turns. You can see that the graph of #math.equation(block: false, alt: "f")[$f$] has three turning points, and the graph of #math.equation(block: false, alt: "g")[$g$] has one turning point. #figure(figph[graphs of two quartic polynomials], alt: "graphs of two quartic polynomials", caption: none) As in Example, both graphs have similar long-term behavior. The #math.equation(block: false, alt: "y")[$y$]-values decrease from #math.equation(block: false, alt: "minus ∞")[$− upright(∞)$] toward zero as #math.equation(block: false, alt: "x")[$x$] increases from #math.equation(block: false, alt: "minus ∞")[$− upright(∞)$], and the #math.equation(block: false, alt: "y")[$y$]-values increase toward #math.equation(block: false, alt: "plus ∞")[$+ upright(∞)$] as #math.equation(block: false, alt: "x")[$x$] increases to #math.equation(block: false, alt: "plus ∞")[$+ upright(∞)$]. This long-term behavior is similar to that of the power function #math.equation(block: false, alt: "y equals x to the power 4")[$y = x^(4)$]. Its graph also starts at the upper left and extends to the upper right. ] ] + Complete the following table of values for #math.equation(block: false, alt: "Q open parenthesis x close parenthesis equals minus x to the power 4 minus x cubed minus 6 x squared plus 2")[$Q ( x ) = − x^(4) − x^(3) − 6 x^(2) + 2$].#figure(table( columns: 6, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]]), [#math.equation(block: false, alt: "y")[$y$]], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], )) #figure(table( columns: 5, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "4")[$4$]]), [#math.equation(block: false, alt: "y")[$y$]], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], )) + Graph #math.equation(block: false, alt: "y equals Q open parenthesis x close parenthesis")[$y = Q ( x )$] in the window #math.equation(block: true, alt: "Xmin, equals minus 5, Xmax equals 5; Ymin, equals minus 15, Ymax equals 10")[$"Xmin" & = − 5 & & "Xmax" = 5 \ "Ymin" & = − 15 & & "Ymax" = 10$] Compare the graph to the graphs in the previous Example: What similarities do you notice? What differences?- The graphs all have four #math.equation(block: false, alt: "x")[$x$]-intercepts and three turning points. \_\_\_\_\_ - The graphs all have long-term behavior like a fourth degree power function, #math.equation(block: false, alt: "y equals a x to the power 4")[$y = a x^(4)$]. \_\_\_\_\_ - The long-term behavior of the graphs in the example is the same as that of #math.equation(block: false, alt: "y equals x to the power 4")[$y = x^(4)$], but the graph here has long-term behavior like #math.equation(block: false, alt: "y equals minus x to the power 4")[$y = − x^(4)$]. \_\_\_\_\_ - The long-term behavior of the graphs in the example is the same as that of #math.equation(block: false, alt: "y equals x to the power 4")[$y = x^(4)$], but the graph here has long-term behavior like #math.equation(block: false, alt: "y equals minus x cubed")[$y = − x^(3)$]. \_\_\_\_\_ #solutionbox[ + #figure(table( columns: 10, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "4")[$4$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "minus 286")[$− 286$]], [#math.equation(block: false, alt: "minus 106")[$− 106$]], [#math.equation(block: false, alt: "minus 30")[$− 30$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "minus 6")[$− 6$]], [#math.equation(block: false, alt: "minus 46")[$− 46$]], [#math.equation(block: false, alt: "minus 160")[$− 160$]], [#math.equation(block: false, alt: "minus 414")[$− 414$]], )) + The graphs all have long-term behavior like a fourth degree power function, #math.equation(block: false, alt: "y equals a x to the power 4")[$y = a x^(4)$]. The long-term behavior of the graphs in the Example is the same as that of #math.equation(block: false, alt: "y equals x to the power 4")[$y = x^(4)$], but the graph here has long-term behavior like #math.equation(block: false, alt: "y equals minus x to the power 4")[$y = − x^(4)$]. ] A graph for part (b): #figure(figph[quartic], alt: "quartic", caption: none) + Complete the following table of values for #math.equation(block: false, alt: "Q open parenthesis x close parenthesis equals minus x to the power 4 minus x cubed minus 6 x squared plus 2")[$" " Q ( x ) = − x^(4) − x^(3) − 6 x^(2) + 2$].#figure(table( columns: 10, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "4")[$4$]]), [#math.equation(block: false, alt: "y")[$y$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], )) + Graph #math.equation(block: false, alt: "y equals Q open parenthesis x close parenthesis")[$y = Q ( x )$] in the window #math.equation(block: true, alt: "Xmin, equals minus 5, Xmax equals 5; Ymin, equals minus 15, Ymax equals 10")[$"Xmin" & = − 5 & & "Xmax" = 5 \ "Ymin" & = − 15 & & "Ymax" = 10$] Compare the graph to the graphs in the previous Example: What similarities do you notice? What differences? #solutionbox[ + #figure(table( columns: 10, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "4")[$4$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "minus 286")[$− 286$]], [#math.equation(block: false, alt: "minus 106")[$− 106$]], [#math.equation(block: false, alt: "minus 30")[$− 30$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "minus 6")[$− 6$]], [#math.equation(block: false, alt: "minus 46")[$− 46$]], [#math.equation(block: false, alt: "minus 160")[$− 160$]], [#math.equation(block: false, alt: "minus 414")[$− 414$]], )) + #figure(figph[quartic], alt: "quartic", caption: none) #linebreak() The graphs all have long-term behavior like a fourth degree power function, #math.equation(block: false, alt: "y equals a x to the power 4")[$y = a x^(4)$]. The long-term behavior of the graphs in the Example is the same as that of #math.equation(block: false, alt: "y equals x to the power 4")[$y = x^(4)$], but the graph here has long-term behavior like #math.equation(block: false, alt: "y equals minus x to the power 4")[$y = − x^(4)$]. ] In Examples and Example, we have seen polynomials of degree 3 and degree 4, whose graphs are illustrated in the next box. In the Homework Problems, you will consider more graphs to help you verify the following observations. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Long-Term Behavior of Polynomial Functions] + A polynomial of odd degree (with positive lead coefficient) has negative #math.equation(block: false, alt: "y")[$y$]-values for large negative #math.equation(block: false, alt: "x")[$x$] and positive #math.equation(block: false, alt: "y")[$y$]-values for large positive #math.equation(block: false, alt: "x")[$x$]. + A polynomial of even degree (with positive lead coefficient) has positive #math.equation(block: false, alt: "y")[$y$]-values for both large positive and large negative #math.equation(block: false, alt: "x")[$x$]. #figure(figph[graphs of a cubic and a quartic], alt: "graphs of a cubic and a quartic", caption: none) ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Another way to describe the long-term behavior of a polynomial graph is to note that for large values of #math.equation(block: false, alt: "vertical bar x vertical bar")[$| x |$], the shape is similar to a power function of the same degree. It is the presence of the lower-degree terms in the polynomial that are responsible for any extra wiggles or turning points in the graph. ] The long-term behavior of a polynomial of degree #math.equation(block: false, alt: "n")[$n$] \_\_\_\_\_ #solutionbox[ is the same as the behavior of #math.equation(block: false, alt: "y equals k x to the power n")[$y = k x^(n)$]. ] The long-term behavior of a polynomial of degree #math.equation(block: false, alt: "n")[$n$] + depends on the number of its zeros. + is the same as the behavior of #math.equation(block: false, alt: "y equals k x to the power n")[$y = k x^(n)$]. + may have a horizontal asymptote. + cannot be determined without a graph. === #math.equation(block: false, alt: "x")[$x$]-Intercepts and the Factor Theorem In Quadratic Functions, we saw that the x-intercepts of a quadratic polynomial, #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals a x squared plus b x plus c")[$f ( x ) = a x^(2) + b x + c$], occur at values of #math.equation(block: false, alt: "x")[$x$] for which #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 0")[$f ( x ) = 0$], that is, at the real-valued solutions of the equation #math.equation(block: false, alt: "a x squared plus b x plus c equals 0")[$a x^(2) + b x + c = 0$]. The same holds true for polynomials of higher degree. Solutions of the equation #math.equation(block: false, alt: "P open parenthesis x close parenthesis equals 0")[$P ( x ) = 0$] are called #strong[zeros] of the polynomial #math.equation(block: false, alt: "P")[$P$]. In Example, we graphed the cubic polynomial #math.equation(block: false, alt: "P open parenthesis x close parenthesis equals x cubed minus 4 x")[$P ( x ) = x^(3) − 4 x$]. Its #math.equation(block: false, alt: "x")[$x$]-intercepts are the solutions of the equation #math.equation(block: false, alt: "x cubed minus 4 x equals 0")[$x^(3) − 4 x = 0$], which we can solve by factoring the polynomial #math.equation(block: false, alt: "P open parenthesis x close parenthesis")[$P ( x )$]. #math.equation(block: true, alt: "x cubed minus 4 x, equals 0; x open parenthesis x minus 2 close parenthesis open parenthesis x plus 2 close parenthesis, equals 0")[$x^(3) − 4 x & = 0 \ x ( x − 2 ) ( x + 2 ) & = 0$] The zeros of #math.equation(block: false, alt: "P")[$P$] are #math.equation(block: false, alt: "0")[$0$], #math.equation(block: false, alt: "2")[$2$], and #math.equation(block: false, alt: "minus 2")[$− 2$]. Each zero of #math.equation(block: false, alt: "P")[$P$] corresponds to a factor of #math.equation(block: false, alt: "P open parenthesis x close parenthesis")[$P ( x )$]. This result suggests the following theorem, which holds for any polynomial #math.equation(block: false, alt: "P")[$P$]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Factor Theorem] Let #math.equation(block: false, alt: "P open parenthesis x close parenthesis")[$P ( x )$] be a polynomial with real number coefficients. Then #math.equation(block: false, alt: "open parenthesis x minus a close parenthesis")[$( x − a )$] is a factor of #math.equation(block: false, alt: "P open parenthesis x close parenthesis")[$P ( x )$] if and only if #math.equation(block: false, alt: "P open parenthesis a close parenthesis equals 0")[$P ( a ) = 0$]. ] Without graphing, how can you tell that the polynomial #math.equation(block: false, alt: "p open parenthesis x close parenthesis equals open parenthesis x squared plus 1 close parenthesis open parenthesis x squared plus 2 close parenthesis")[$p ( x ) = ( x^(2) + 1 ) ( x^(2) + 2 )$] has no #math.equation(block: false, alt: "x")[$x$]-intercepts? \_\_\_\_\_ #solutionbox[ It cannot be factored further. Because the two quadratic factors are both positive for all #math.equation(block: false, alt: "x")[$x$], their product is always positive. ] Without graphing, how can you tell that the polynomial #math.equation(block: false, alt: "p open parenthesis x close parenthesis equals open parenthesis x squared plus 1 close parenthesis open parenthesis x squared plus 2 close parenthesis")[$" " p ( x ) = ( x^(2) + 1 ) ( x^(2) + 2 ) " "$] has no #math.equation(block: false, alt: "x")[$x$]-intercepts? + It is quartic. + All of its coefficients are positive. + It cannot be factored further. + It has no turning points. The factor theorem follows from the division algorithm for polynomials. (See the Polynomials and Factoring section in Appendix Algebra Skills Refresher to review polynomial division.) We will consider both of these results in more detail in the Homework problems. The graph of which type of polynomial must have an #math.equation(block: false, alt: "x")[$x$]-intercept? \_\_\_\_\_ #solutionbox[ The graph of any cubic polynomial must have at least one #math.equation(block: false, alt: "x")[$x$]-intercept. ] The graph of which type of polynomial must have an #math.equation(block: false, alt: "x")[$x$]-intercept? + Quadratic + Cubic + Quartic + None of these Because a polynomial function #math.equation(block: false, alt: "P")[$P$] of degree #math.equation(block: false, alt: "n")[$n$] can have at most #math.equation(block: false, alt: "n")[$n$] linear factors of the form #math.equation(block: false, alt: "open parenthesis x minus a close parenthesis")[$( x − a )$], it follows that #math.equation(block: false, alt: "P")[$P$] can have at most #math.equation(block: false, alt: "n")[$n$] distinct zeros. Another way of saying this is that if #math.equation(block: false, alt: "P open parenthesis x close parenthesis")[$P ( x )$] is a polynomial of #math.equation(block: false, alt: "n")[$n$]th degree, the equation #math.equation(block: false, alt: "P open parenthesis x close parenthesis equals 0")[$P ( x ) = 0$] can have at most #math.equation(block: false, alt: "n")[$n$] distinct solutions, some of which may be complex numbers. (We consider complex numbers in Complex Numbers.) Because only real-valued solutions appear on the graph as #math.equation(block: false, alt: "x")[$x$]-intercepts, we have the following corollary to the factor theorem. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[#math.equation(block: false, alt: "x")[$x$]-Intercepts of Polynomials] A polynomial of degree #math.equation(block: false, alt: "n")[$n$] can have at most #math.equation(block: false, alt: "n")[$n$] #math.equation(block: false, alt: "x")[$x$]-intercepts. ] If some of the zeros of #math.equation(block: false, alt: "P")[$P$] are complex numbers, they will not appear on the graph, so a polynomial of degree #math.equation(block: false, alt: "n")[$n$] #emph[may have fewer] than #math.equation(block: false, alt: "n")[$n$] #math.equation(block: false, alt: "x")[$x$]-intercepts. #examplebox("Example 3")[][ Find the real-valued zeros of each of the following polynomials, and list the #math.equation(block: false, alt: "x")[$x$]-intercepts of its graph. + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed plus 6 x squared plus 9 x")[$f ( x ) = x^(3) + 6 x^(2) + 9 x$] + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x to the power 4 minus 3 x squared minus 4")[$g ( x ) = x^(4) − 3 x^(2) − 4$] #solutionbox[ + Factor the polynomial to obtain #math.equation(block: true, alt: "f open parenthesis x close parenthesis, equals x open parenthesis x squared plus 6 x plus 9 close parenthesis; equals x open parenthesis x plus 3 close parenthesis open parenthesis x plus 3 close parenthesis")[$f ( x ) & = x ( x^(2) + 6 x + 9 ) \ & = x ( x + 3 ) ( x + 3 )$] By the factor theorem, the zeros of #math.equation(block: false, alt: "f")[$f$] are #math.equation(block: false, alt: "0")[$0$], #math.equation(block: false, alt: "minus 3")[$− 3$], and #math.equation(block: false, alt: "minus 3")[$− 3$]. (We say that #math.equation(block: false, alt: "f")[$f$] has a zero #emph[of multiplicity two] at #math.equation(block: false, alt: "minus 3")[$− 3$].) Because all of these are real numbers, all will appear as #math.equation(block: false, alt: "x")[$x$]-intercepts on the graph. Thus, the #math.equation(block: false, alt: "x")[$x$]-intercepts occur at #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] and at #math.equation(block: false, alt: "open parenthesis minus 3 , 0 close parenthesis")[$( − 3 , 0 )$]. + Factor the polynomial to obtain #math.equation(block: true, alt: "g open parenthesis x close parenthesis, equals open parenthesis x squared minus 4 close parenthesis open parenthesis x squared plus 1 close parenthesis; equals open parenthesis x minus 2 close parenthesis open parenthesis x plus 2 close parenthesis open parenthesis x squared plus 1 close parenthesis")[$g ( x ) & = ( x^(2) − 4 ) ( x^(2) + 1 ) \ & = ( x − 2 ) ( x + 2 ) ( x^(2) + 1 )$] Because #math.equation(block: false, alt: "x squared plus 1")[$x^(2) + 1$] cannot be factored in real numbers, the graph has only two #math.equation(block: false, alt: "x")[$x$]-intercepts, at #math.equation(block: false, alt: "open parenthesis minus 2 , 0 close parenthesis")[$( − 2 , 0 )$] and #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$]. The graphs of both polynomials are shown below.#figure(figph[a cubic with two zeros and a quartic with two zeros], alt: "a cubic with two zeros and a quartic with two zeros", caption: none) ] ] + Find the real-valued zeros of #math.equation(block: false, alt: "P open parenthesis x close parenthesis equals minus x to the power 4 plus x cubed plus 2 x squared")[$P ( x ) = − x^(4) + x^(3) + 2 x^(2)$] by factoring. #linebreak() \_\_\_\_\_ Separate different zeros with a comma. + Sketch a rough graph of #math.equation(block: false, alt: "y equals P open parenthesis x close parenthesis")[$y = P ( x )$] by hand. #solutionbox[ + #math.equation(block: false, alt: "minus 1 , 0 , 2")[$− 1 , 0 , 2$] + A graph is below. ] Graph for part (b): #figure(figph[quartic], alt: "quartic", caption: none) + Find the real-valued zeros of #math.equation(block: false, alt: "P open parenthesis x close parenthesis equals minus x to the power 4 plus x cubed plus 2 x squared")[$" " P ( x ) = − x^(4) + x^(3) + 2 x^(2) " "$] by factoring. + Sketch a rough graph of #math.equation(block: false, alt: "y equals P open parenthesis x close parenthesis")[$y = P ( x )$] by hand. #solutionbox[ + #math.equation(block: false, alt: "minus 1 , 0 , 2")[$− 1 , 0 , 2$] + #figure(figph[quartic], alt: "quartic", caption: none) ] How may the graph of a polynomial differ from the graph of a power function of the same degree? \_\_\_\_\_ How may the graph of a polynomial differ from the graph of a power function of the same degree? === Zeros of Multiplicity Two or Three The appearance of the graph near an #math.equation(block: false, alt: "x")[$x$]-intercept is determined by the multiplicity of the zero there. - Both real zeros of the polynomial #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x to the power 4 minus 3 x squared minus 4")[$g ( x ) = x^(4) − 3 x^(2) − 4$] in Exampleb are of multiplicity one, and the graph #emph[crosses] the #math.equation(block: false, alt: "x")[$x$]-axis at each intercept. - The polynomial #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed plus 6 x squared plus 9 x")[$f ( x ) = x^(3) + 6 x^(2) + 9 x$] in Examplea has a zero of multiplicity two at #math.equation(block: false, alt: "x equals minus 3")[$x = − 3$]. The graph of #math.equation(block: false, alt: "f")[$f$] just #emph[touches] the #math.equation(block: false, alt: "x")[$x$]-axis and then reverses direction without crossing the axis. To understand what happens in general, compare the graphs of the three polynomials below. #figure(figph[plots of a linear, quadratic, and cubic, each with a single zero], alt: "plots of a linear, quadratic, and cubic, each with a single zero", caption: none) + In figure (a), #math.equation(block: false, alt: "L open parenthesis x close parenthesis equals x minus 2")[$L ( x ) = x − 2$] has a zero #emph[of multiplicity one] at #math.equation(block: false, alt: "x equals 2")[$x = 2$], and its graph crosses the #math.equation(block: false, alt: "x")[$x$]-axis there. + In figure (b), #math.equation(block: false, alt: "Q open parenthesis x close parenthesis equals open parenthesis x minus 2 close parenthesis squared")[$Q ( x ) = ( x − 2 )^(2)$] has a zero #emph[of multiplicity two] at #math.equation(block: false, alt: "x equals 2")[$x = 2$], and its graph touches the #math.equation(block: false, alt: "x")[$x$]-axis there but changes direction without crossing. + In figure (c), #math.equation(block: false, alt: "C open parenthesis x close parenthesis equals open parenthesis x minus 2 close parenthesis cubed")[$C ( x ) = ( x − 2 )^(3)$] has a zero #emph[of multiplicity three] at #math.equation(block: false, alt: "x equals 2")[$x = 2$]. In this case, the graph makes an S-shaped curve at the intercept, like the graph of #math.equation(block: false, alt: "y equals x cubed")[$y = x^(3)$]. Near its #math.equation(block: false, alt: "x")[$x$]-intercepts, the graph of a polynomial takes one of the characteristic shapes illustrated above. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Although we will not consider zeros of multiplicity greater than three, they correspond to similar behavior in the graph: - At a zero of odd multiplicity, the graph has an inflection point at the intercept; its graph makes an S-shaped curve. - At a zero of even multiplicity, the graph has a turning point; it changes direction without crossing the #math.equation(block: false, alt: "x")[$x$]-axis. ] #examplebox("Example 4")[][ Graph the polynomial #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals open parenthesis x plus 2 close parenthesis cubed open parenthesis x minus 1 close parenthesis open parenthesis x minus 3 close parenthesis squared")[$f ( x ) = ( x + 2 )^(3) ( x − 1 ) ( x − 3 )^(2)$] #solutionbox[ The polynomial has degree six, an even number, so its graph starts at the upper left and extends to the upper right. Its #math.equation(block: false, alt: "y")[$y$]-intercept is #math.equation(block: true, alt: "f open parenthesis 0 close parenthesis equals open parenthesis 2 close parenthesis cubed open parenthesis minus 1 close parenthesis open parenthesis minus 3 close parenthesis squared equals minus 72")[$f ( 0 ) = ( 2 )^(3) ( − 1 ) ( − 3 )^(2) = − 72$] #math.equation(block: false, alt: "f")[$f$] has a zero of multiplicity three at #math.equation(block: false, alt: "x equals minus 2")[$x = − 2$], a zero of multiplicity one at #math.equation(block: false, alt: "x equals 1")[$x = 1$], and a zero of multiplicity two at #math.equation(block: false, alt: "x equals 3")[$x = 3$]. The graph has an S-shaped curve at #math.equation(block: false, alt: "x equals minus 2")[$x = − 2$], crosses the #math.equation(block: false, alt: "x")[$x$]-axis at #math.equation(block: false, alt: "x equals 1")[$x = 1$], touches the #math.equation(block: false, alt: "x")[$x$]-axis at #math.equation(block: false, alt: "x equals 3")[$x = 3$], and then changes direction, as shown below. #figure(figph[graph of sixth degree polynomial with zero of multiplicity 3, another of multiplicity two. and a zero of multiplicity 1], alt: "graph of sixth degree polynomial with zero of multiplicity 3, another of multiplicity two. and a zero of multiplicity 1", caption: none) ] ] Sketch a rough graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals open parenthesis x plus 3 close parenthesis open parenthesis x minus 1 close parenthesis squared")[$f ( x ) = ( x + 3 ) ( x − 1 )^(2)$] by hand. Label the #math.equation(block: false, alt: "x")[$x$]- and #math.equation(block: false, alt: "y")[$y$]-intercepts. #solutionbox[ #math.equation(block: false, alt: "f")[$f$] has degree three with a positive coefficient of #math.equation(block: false, alt: "x cubed")[$x^(3)$], so its graph starts at the lower left and extends to the upper right. The graph crosses the #math.equation(block: false, alt: "x")[$x$]-axis at #math.equation(block: false, alt: "x equals minus 3")[$x = − 3$], touches the #math.equation(block: false, alt: "x")[$x$]-axis at the double zero #math.equation(block: false, alt: "x equals 1")[$x = 1$] and then changes direction, as shown in the graph below. ] #figure(figph[cubic], alt: "cubic", caption: none) Sketch a rough graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals open parenthesis x plus 3 close parenthesis open parenthesis x minus 1 close parenthesis squared")[$" " f ( x ) = ( x + 3 ) ( x − 1 )^(2) " "$] by hand. Label the #math.equation(block: false, alt: "x")[$x$]- and #math.equation(block: false, alt: "y")[$y$]-intercepts. #solutionbox[ #math.equation(block: false, alt: "f")[$f$] has degree three with a positive coefficient of #math.equation(block: false, alt: "x cubed")[$x^(3)$], so its graph starts at the lower left and extends to the upper right. The graph crosses the #math.equation(block: false, alt: "x")[$x$]-axis at #math.equation(block: false, alt: "x equals minus 3")[$x = − 3$], touches the #math.equation(block: false, alt: "x")[$x$]-axis at the double zero #math.equation(block: false, alt: "x equals 1")[$x = 1$] and then changes direction, as shown in the graph below. #figure(figph[cubic], alt: "cubic", caption: none) ] Explain how the multiplicity of a zero affects the shape of the graph near that zero. \_\_\_\_\_ Explain how the multiplicity of a zero affects the shape of the graph near that zero. === Section Summary ==== Vocabulary Look up the definitions of new terms in the Glossary. - Degree - Multiplicity - Inflection point - Quartic - Turning point - Cubic - Corollary - Zero ==== CONCEPTS + The graphs of all polynomials are smooth curves without breaks or holes. + The graph of a polynomial of degree #math.equation(block: false, alt: "n")[$n$] (with positive leading coefficient) has the same long-term behavior as the power function of the same degree. + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Factor Theorem] Let #math.equation(block: false, alt: "P open parenthesis x close parenthesis")[$P ( x )$] be a polynomial with real number coefficients. Then #math.equation(block: false, alt: "open parenthesis x minus a close parenthesis")[$( x − a )$] is a factor of #math.equation(block: false, alt: "P open parenthesis x close parenthesis")[$P ( x )$] if and only if #math.equation(block: false, alt: "P open parenthesis a close parenthesis equals 0")[$P ( a ) = 0$]. ] + A polynomial of degree #math.equation(block: false, alt: "n")[$n$] can have at most #math.equation(block: false, alt: "n")[$n$] #math.equation(block: false, alt: "x")[$x$]-intercepts. + At a zero of multiplicity two, the graph of a polynomial has a turning point. At a zero of multiplicity three, the graph of a polynomial has an inflection point. ==== STUDY QUESTIONS + Describe the graphs of polynomials of degrees #math.equation(block: false, alt: "0")[$0$], #math.equation(block: false, alt: "1")[$1$], and #math.equation(block: false, alt: "2")[$2$]. + What does the degree of a polynomial tell you about its long-term behavior? + What is a zero of a polynomial? + How are zeros related to the factors of a polynomial? + What do the zeros tell you about the graph of a polynomial? + Explain the difference between a turning point and an inflection point. ==== SKILLS Practice each skill in the Homework problems listed. + Identify #math.equation(block: false, alt: "x")[$x$]-intercepts, turning points, and inflection points: \#1–8, 11–18 + Use a graph to factor a polynomial: \#21–28 + Sketch the graph of a polynomial: \#29–46 + Find a possible formula for a polynomial whose graph is shown: \#47–52 + Graph translations of polynomials: \#53–56 === Homework 7.2 In Problems 1–8, use your calculator to graph each cubic polynomial. + Describe the long-term behavior of each graph. How does this behavior compare to that of the basic cubic? How does the sign of the lead coefficient affect the graph? + How many #math.equation(block: false, alt: "x")[$x$]-intercepts does each graph have? How many turning points? How many inflection points? #math.equation(block: true, alt: "y equals x cubed plus 4")[$y = x^(3) + 4$] #figure(figph[cubic], alt: "cubic", caption: none) + The end behavior is the same as for the basic cubic because the lead coefficient is positive. + There is one #math.equation(block: false, alt: "x")[$x$]-intercept, no turning points, one inflection point. #math.equation(block: true, alt: "y equals x cubed minus 8")[$y = x^(3) − 8$] #math.equation(block: true, alt: "y equals minus 2 minus 0.05 x cubed")[$y = − 2 − 0.05 x^(3)$] #figure(figph[cubic], alt: "cubic", caption: none) + The end behavior is the opposite to the basic cubic (the graph starts in the upper left and extends to the lower right) because the lead coefficient is negative. + There is one #math.equation(block: false, alt: "x")[$x$]-intercept, no turning points, one inflection point. #math.equation(block: true, alt: "y equals 5 minus 0.02 x cubed")[$y = 5 − 0.02 x^(3)$] #math.equation(block: true, alt: "y equals x cubed minus 3 x")[$y = x^(3) − 3 x$] #figure(figph[cubic], alt: "cubic", caption: none) + The end behavior is the same as for the basic cubic because the lead coefficient is positive. + There are three #math.equation(block: false, alt: "x")[$x$]-intercepts, two turning points, one inflection point. #math.equation(block: true, alt: "y equals 9 x minus x cubed")[$y = 9 x − x^(3)$] #math.equation(block: true, alt: "y equals x cubed plus 5 x squared minus 4 x minus 20")[$y = x^(3) + 5 x^(2) − 4 x − 20$] #figure(figph[cubic], alt: "cubic", caption: none) + The end behavior is the same as for the basic cubic because the lead coefficient is positive. + There are three #math.equation(block: false, alt: "x")[$x$]-intercepts, two turning points, one inflection point. #math.equation(block: true, alt: "y equals minus x cubed minus 2 x squared plus 5 x plus 6")[$y = − x^(3) − 2 x^(2) + 5 x + 6$] For Problems 9–10, use a graphing utility to graph each cubic polynomial. Which graphs are the same? + #math.equation(block: false, alt: "y equals x cubed minus 2")[$y = x^(3) − 2$] + #math.equation(block: false, alt: "y equals open parenthesis x minus 2 close parenthesis cubed")[$y = ( x − 2 )^(3)$] + #math.equation(block: false, alt: "y equals x cubed minus 6 x squared plus 12 x minus 8")[$y = x^(3) − 6 x^(2) + 12 x − 8$] + #figure(figph[cubic], alt: "cubic", caption: none) + #figure(figph[cubic], alt: "cubic", caption: none) + #figure(figph[cubic], alt: "cubic", caption: none) (b) and (c) are the same. + #math.equation(block: false, alt: "y equals x cubed plus 3")[$y = x^(3) + 3$] + #math.equation(block: false, alt: "y equals open parenthesis x plus 3 close parenthesis cubed")[$y = ( x + 3 )^(3)$] + #math.equation(block: false, alt: "y equals x cubed plus 9 x squared plus 27 x plus 27")[$y = x^(3) + 9 x^(2) + 27 x + 27$] In Problems 11–18, use your calculator to graph each quartic polynomial. + Describe the long-term behavior of each graph. How does this behavior compare to that of the basic quartic? How does the sign of the lead coefficient affect the graph? + How many #math.equation(block: false, alt: "x")[$x$]-intercepts does each graph have? How many turning points? How many inflection points? #math.equation(block: true, alt: "y equals 0.5 x to the power 4 minus 4")[$y = 0.5 x^(4) − 4$] #figure(figph[quartic], alt: "quartic", caption: none) + The end behavior is the same as for the basic quartic because the lead coefficient is positive. + There are two #math.equation(block: false, alt: "x")[$x$]-intercepts, one turning point, no inflection point. #math.equation(block: true, alt: "y equals 0.3 x to the power 4 plus 1")[$y = 0.3 x^(4) + 1$] #math.equation(block: true, alt: "y equals minus x to the power 4 plus 6 x squared minus 10")[$y = − x^(4) + 6 x^(2) − 10$] #figure(figph[quartic], alt: "quartic", caption: none) + The end behavior is the opposite of the basic quartic (the graph starts in the lower left and ends in the lower right) because the lead coefficient is negative. + There are no #math.equation(block: false, alt: "x")[$x$]-intercepts, three turning points, two inflection points. #math.equation(block: true, alt: "y equals x to the power 4 minus 8 x squared minus 8")[$y = x^(4) − 8 x^(2) − 8$] #math.equation(block: true, alt: "y equals x to the power 4 minus 3 x cubed")[$y = x^(4) − 3 x^(3)$] #figure(figph[quartic], alt: "quartic", caption: none) + The end behavior is the same as for the basic quartic because the lead coefficient is positive. + There are two #math.equation(block: false, alt: "x")[$x$]-intercepts, one turning point, two inflection points. #math.equation(block: true, alt: "y equals minus x to the power 4 minus 4 x cubed")[$y = − x^(4) − 4 x^(3)$] #math.equation(block: true, alt: "y equals minus x to the power 4 minus x cubed minus 2")[$y = − x^(4) − x^(3) − 2$] #figure(figph[quartic], alt: "quartic", caption: none) + The end behavior is the opposite of the basic quartic (the graph starts in the lower left and ends in the lower right) because the lead coefficient is negative. + There are no #math.equation(block: false, alt: "x")[$x$]-intercepts, one turning point, two inflection points. #math.equation(block: true, alt: "y equals x to the power 4 plus 2 x cubed plus 4 x squared plus 10")[$y = x^(4) + 2 x^(3) + 4 x^(2) + 10$] From your answers to Problems 1–8, what you can conclude about the graphs of cubic polynomials? Consider the long-term behavior, #math.equation(block: false, alt: "x")[$x$]-intercepts, turning points, and inflection points. The graph of a cubic polynomial with a positive lead coefficient will have the same end behavior as the basic cubic, and a cubic with a negative lead coefficient will have the opposite end behavior. Each graph of a cubic polynomial has one, two, or three #math.equation(block: false, alt: "x")[$x$]-intercepts, it has two, one or no turning point, and it has exactly one inflection point. From your answers to Problems 11–18, what you can conclude about the graphs of quartic polynomials? Consider the long-term behavior, #math.equation(block: false, alt: "x")[$x$]-intercepts, turning points, and inflection points. For Problems 21–28, + Use your calculator to graph each polynomial and locate the #math.equation(block: false, alt: "x")[$x$]-intercepts. Set #math.equation(block: false, alt: "Xmin equals minus 4.7")[$"Xmin" = − 4.7$], #math.equation(block: false, alt: "Xmax equals 4.7")[$"Xmax" = 4.7$], and adjust #math.equation(block: false, alt: "Ymin")[$"Ymin"$] and #math.equation(block: false, alt: "Ymax")[$"Ymax"$] to get a good graph. + Write the polynomial in factored form. + Expand the factored form of the polynomial (that is, multiply the factors together). Do you get the original polynomial? #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals x cubed minus 7 x minus 6")[$P ( x ) = x^(3) − 7 x − 6$] + #figure(figph[cubic], alt: "cubic", caption: none) #linebreak() #math.equation(block: false, alt: "open parenthesis minus 2 , 0 close parenthesis , open parenthesis minus 1 , 0 close parenthesis , open parenthesis 3 , 0 close parenthesis")[$( − 2 , 0 ) , ( − 1 , 0 ) , ( 3 , 0 )$] + #math.equation(block: false, alt: "P open parenthesis x close parenthesis equals open parenthesis x plus 2 close parenthesis open parenthesis x plus 1 close parenthesis open parenthesis x minus 3 close parenthesis")[$P ( x ) = ( x + 2 ) ( x + 1 ) ( x − 3 )$] + Yes #math.equation(block: true, alt: "Q open parenthesis x close parenthesis equals x cubed plus 3 x squared minus x minus 3")[$Q ( x ) = x^(3) + 3 x^(2) − x − 3$] #math.equation(block: true, alt: "R open parenthesis x close parenthesis equals x to the power 4 minus x cubed minus 4 x squared plus 4 x")[$R ( x ) = x^(4) − x^(3) − 4 x^(2) + 4 x$] + #figure(figph[quartic], alt: "quartic", caption: none) #linebreak() #math.equation(block: false, alt: "open parenthesis minus 2 , 0 close parenthesis , open parenthesis 0 , 0 close parenthesis , open parenthesis 1 , 0 close parenthesis , open parenthesis 2 , 0 close parenthesis")[$( − 2 , 0 ) , ( 0 , 0 ) , ( 1 , 0 ) , ( 2 , 0 )$] + #math.equation(block: false, alt: "R open parenthesis x close parenthesis equals open parenthesis x plus 2 close parenthesis open parenthesis x close parenthesis open parenthesis x minus 1 close parenthesis open parenthesis x minus 2 close parenthesis")[$R ( x ) = ( x + 2 ) ( x ) ( x − 1 ) ( x − 2 )$] + Yes #math.equation(block: true, alt: "S open parenthesis x close parenthesis equals x to the power 4 plus 3 x cubed minus x squared minus 3 x")[$S ( x ) = x^(4) + 3 x^(3) − x^(2) − 3 x$] #math.equation(block: true, alt: "p open parenthesis x close parenthesis equals x cubed minus 3 x squared minus 6 x plus 8")[$p ( x ) = x^(3) − 3 x^(2) − 6 x + 8$] + #figure(figph[cubic], alt: "cubic", caption: none) #linebreak() #math.equation(block: false, alt: "open parenthesis minus 2 , 0 close parenthesis , open parenthesis 1 , 0 close parenthesis , open parenthesis 4 , 0 close parenthesis")[$( − 2 , 0 ) , ( 1 , 0 ) , ( 4 , 0 )$] + #math.equation(block: false, alt: "p open parenthesis x close parenthesis equals open parenthesis x plus 2 close parenthesis open parenthesis x minus 1 close parenthesis open parenthesis x minus 4 close parenthesis")[$p ( x ) = ( x + 2 ) ( x − 1 ) ( x − 4 )$] + Yes #math.equation(block: true, alt: "q open parenthesis x close parenthesis equals x cubed plus 6 x squared minus x minus 30")[$q ( x ) = x^(3) + 6 x^(2) − x − 30$] #math.equation(block: true, alt: "r open parenthesis x close parenthesis equals x to the power 4 minus x cubed minus 10 x squared plus 4 x plus 24")[$r ( x ) = x^(4) − x^(3) − 10 x^(2) + 4 x + 24$] + #figure(figph[quartic], alt: "quartic", caption: none) #linebreak() #math.equation(block: false, alt: "open parenthesis minus 2 , 0 close parenthesis , open parenthesis 2 , 0 close parenthesis , open parenthesis 3 , 0 close parenthesis")[$( − 2 , 0 ) , ( 2 , 0 ) , ( 3 , 0 )$] + #math.equation(block: false, alt: "r open parenthesis x close parenthesis equals open parenthesis x plus 2 close parenthesis squared open parenthesis x minus 2 close parenthesis open parenthesis x minus 3 close parenthesis")[$r ( x ) = ( x + 2 )^(2) ( x − 2 ) ( x − 3 )$] + Yes #math.equation(block: true, alt: "s open parenthesis x close parenthesis equals x to the power 4 minus x cubed minus 12 x squared minus 4 x plus 16")[$s ( x ) = x^(4) − x^(3) − 12 x^(2) − 4 x + 16$] For Problems 29–36, sketch a rough graph of each polynomial function by hand, paying attention to the shape of the graph near each #math.equation(block: false, alt: "x")[$x$]-intercept. Check by graphing with a calculator. #math.equation(block: true, alt: "q open parenthesis x close parenthesis equals open parenthesis x plus 4 close parenthesis open parenthesis x plus 1 close parenthesis open parenthesis x minus 1 close parenthesis")[$q ( x ) = ( x + 4 ) ( x + 1 ) ( x − 1 )$] #figure(figph[cubic], alt: "cubic", caption: none) #math.equation(block: true, alt: "p open parenthesis x close parenthesis equals x open parenthesis x plus 2 close parenthesis open parenthesis x plus 4 close parenthesis")[$p ( x ) = x ( x + 2 ) ( x + 4 )$] #math.equation(block: true, alt: "G open parenthesis x close parenthesis equals open parenthesis x minus 2 close parenthesis squared open parenthesis x plus 2 close parenthesis squared")[$G ( x ) = ( x − 2 )^(2) ( x + 2 )^(2)$] #figure(figph[quartic], alt: "quartic", caption: none) #math.equation(block: true, alt: "F open parenthesis x close parenthesis equals open parenthesis x minus 1 close parenthesis squared open parenthesis x minus 3 close parenthesis squared")[$F ( x ) = ( x − 1 )^(2) ( x − 3 )^(2)$] #math.equation(block: true, alt: "h open parenthesis x close parenthesis equals x cubed open parenthesis x plus 2 close parenthesis open parenthesis x minus 2 close parenthesis")[$h ( x ) = x^(3) ( x + 2 ) ( x − 2 )$] #figure(figph[quintic], alt: "quintic", caption: none) #math.equation(block: true, alt: "H open parenthesis x close parenthesis equals open parenthesis x plus 1 close parenthesis cubed open parenthesis x minus 2 close parenthesis squared")[$H ( x ) = ( x + 1 )^(3) ( x − 2 )^(2)$] #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals open parenthesis x plus 4 close parenthesis squared open parenthesis x plus 1 close parenthesis squared open parenthesis x minus 1 close parenthesis squared")[$P ( x ) = ( x + 4 )^(2) ( x + 1 )^(2) ( x − 1 )^(2)$] #figure(figph[degree six], alt: "degree six", caption: none) \> #math.equation(block: true, alt: "Q open parenthesis x close parenthesis equals x squared open parenthesis x minus 5 close parenthesis open parenthesis x minus 1 close parenthesis squared open parenthesis x plus 2 close parenthesis")[$Q ( x ) = x^(2) ( x − 5 ) ( x − 1 )^(2) ( x + 2 )$] For Problems 37–46, + Find the zeros of each polynomial by factoring. + Sketch a rough graph by hand. #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals x to the power 4 plus 4 x squared")[$P ( x ) = x^(4) + 4 x^(2)$] + #math.equation(block: false, alt: "0")[$0$] (multiplicity 2) + #figure(figph[quartic], alt: "quartic", caption: none) #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals x cubed plus 3 x")[$P ( x ) = x^(3) + 3 x$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x to the power 4 plus 4 x cubed plus 4 x squared")[$f ( x ) = x^(4) + 4 x^(3) + 4 x^(2)$] + #math.equation(block: false, alt: "0")[$0$] (multiplicity 2), #math.equation(block: false, alt: "2")[$2$] (multiplicity 2) + #figure(figph[quartic], alt: "quartic", caption: none) #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals x to the power 4 plus 4 x cubed plus 3 x squared")[$g ( x ) = x^(4) + 4 x^(3) + 3 x^(2)$] #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals 4 x minus x cubed")[$g ( x ) = 4 x − x^(3)$] + #math.equation(block: false, alt: "0 , plus or minus 2")[$0 , ± 2$] + #figure(figph[cubic], alt: "cubic", caption: none) #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 8 x minus x to the power 4")[$f ( x ) = 8 x − x^(4)$] #math.equation(block: true, alt: "K open parenthesis x close parenthesis equals x to the power 4 minus 10 x squared plus 16")[$K ( x ) = x^(4) − 10 x^(2) + 16$] + #math.equation(block: false, alt: "plus or minus the square root of 2 , plus or minus the square root of 8")[$± sqrt(2) , ± sqrt(8)$] + #figure(figph[quartic], alt: "quartic", caption: none) #math.equation(block: true, alt: "m open parenthesis x close parenthesis equals x to the power 4 minus 15 x squared plus 36")[$m ( x ) = x^(4) − 15 x^(2) + 36$] #math.equation(block: true, alt: "r open parenthesis x close parenthesis equals open parenthesis x squared minus 1 close parenthesis open parenthesis x plus 3 close parenthesis squared")[$r ( x ) = ( x^(2) − 1 ) ( x + 3 )^(2)$] + #math.equation(block: false, alt: "plus or minus 1 , minus 3")[$± 1 , − 3$] (multiplicity 2) + #figure(figph[quartic], alt: "quartic", caption: none) #math.equation(block: true, alt: "s open parenthesis x close parenthesis equals open parenthesis x squared minus 9 close parenthesis open parenthesis x minus 1 close parenthesis squared")[$s ( x ) = ( x^(2) − 9 ) ( x − 1 )^(2)$] For Problems 47–52, find a possible equation for the polynomial whose graph is shown. #figure(figph[cubic], alt: "cubic", caption: none) #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals open parenthesis x plus 2 close parenthesis open parenthesis x minus 1 close parenthesis open parenthesis x minus 4 close parenthesis")[$P ( x ) = ( x + 2 ) ( x − 1 ) ( x − 4 )$] #figure(figph[cubic], alt: "cubic", caption: none) #figure(figph[cubic], alt: "cubic", caption: none) #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals open parenthesis x plus 3 close parenthesis squared open parenthesis x minus 2 close parenthesis")[$P ( x ) = ( x + 3 )^(2) ( x − 2 )$] #figure(figph[cubic], alt: "cubic", caption: none) #figure(figph[quartic], alt: "quartic", caption: none) #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals open parenthesis x minus 2 close parenthesis cubed open parenthesis x plus 2 close parenthesis")[$P ( x ) = ( x − 2 )^(3) ( x + 2 )$] #figure(figph[quartic], alt: "quartic", caption: none) For Problems 53–56, write the formula for each function in parts (a) through (d) and graph with a calculator. Describe how the graph differs from the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$]. #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x cubed minus 4 x")[$f ( x ) = x^(3) − 4 x$] + #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis plus 3")[$y = f ( x ) + 3$] + #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis minus 5")[$y = f ( x ) − 5$] + #math.equation(block: false, alt: "y equals f open parenthesis x minus 2 close parenthesis")[$y = f ( x − 2 )$] + #math.equation(block: false, alt: "y equals f open parenthesis x plus 3 close parenthesis")[$y = f ( x + 3 )$] + #math.equation(block: false, alt: "y equals x cubed minus 4 x plus 3")[$y = x^(3) − 4 x + 3$]; The graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] shifted #math.equation(block: false, alt: "3")[$3$] units up.#figure(figph[cubics], alt: "cubics", caption: none) + #math.equation(block: false, alt: "y equals x cubed minus 4 x minus 5")[$y = x^(3) − 4 x − 5$]; The graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] shifted #math.equation(block: false, alt: "5")[$5$] units down.#figure(figph[cubics], alt: "cubics", caption: none) + #math.equation(block: false, alt: "y equals open parenthesis x minus 2 close parenthesis cubed minus 4 open parenthesis x minus 2 close parenthesis")[$y = ( x − 2 )^(3) − 4 ( x − 2 )$]; The graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] shifted #math.equation(block: false, alt: "2")[$2$] units right.#figure(figph[cubics], alt: "cubics", caption: none) + #math.equation(block: false, alt: "y equals open parenthesis x plus 3 close parenthesis cubed minus 4 open parenthesis x plus 3 close parenthesis")[$y = ( x + 3 )^(3) − 4 ( x + 3 )$]; The graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] shifted #math.equation(block: false, alt: "3")[$3$] units left.#figure(figph[cubics], alt: "cubics", caption: none) #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x cubed minus x squared plus x minus 1")[$f ( x ) = x^(3) − x^(2) + x − 1$] + #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis plus 4")[$y = f ( x ) + 4$] + #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis minus 4")[$y = f ( x ) − 4$] + #math.equation(block: false, alt: "y equals f open parenthesis x minus 3 close parenthesis")[$y = f ( x − 3 )$] + #math.equation(block: false, alt: "y equals f open parenthesis x plus 5 close parenthesis")[$y = f ( x + 5 )$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x to the power 4 minus 4 x squared")[$f ( x ) = x^(4) − 4 x^(2)$] + #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis plus 6")[$y = f ( x ) + 6$] + #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis minus 2")[$y = f ( x ) − 2$] + #math.equation(block: false, alt: "y equals f open parenthesis x minus 1 close parenthesis")[$y = f ( x − 1 )$] + #math.equation(block: false, alt: "y equals f open parenthesis x plus 2 close parenthesis")[$y = f ( x + 2 )$] + #math.equation(block: false, alt: "y equals x to the power 4 minus 4 x squared plus 6")[$y = x^(4) − 4 x^(2) + 6$]; The graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] shifted #math.equation(block: false, alt: "6")[$6$] units up.#figure(figph[quartics], alt: "quartics", caption: none) + #math.equation(block: false, alt: "y equals x to the power 4 minus 4 x squared minus 2")[$y = x^(4) − 4 x^(2) − 2$]; The graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] shifted #math.equation(block: false, alt: "2")[$2$] units down.#figure(figph[quartics], alt: "quartics", caption: none) + #math.equation(block: false, alt: "y equals open parenthesis x minus 1 close parenthesis to the power 4 minus 4 open parenthesis x minus 1 close parenthesis squared")[$y = ( x − 1 )^(4) − 4 ( x − 1 )^(2)$]; The graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] shifted #math.equation(block: false, alt: "1")[$1$] unit right.#figure(figph[quartics], alt: "quartics", caption: none) + #math.equation(block: false, alt: "y equals open parenthesis x plus 2 close parenthesis to the power 4 minus 4 open parenthesis x plus 2 close parenthesis squared")[$y = ( x + 2 )^(4) − 4 ( x + 2 )^(2)$]; The graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] shifted #math.equation(block: false, alt: "2")[$2$] units left.#figure(figph[quartics], alt: "quartics", caption: none) #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x to the power 4 plus 3 x cubed")[$f ( x ) = x^(4) + 3 x^(3)$] + #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis plus 5")[$y = f ( x ) + 5$] + #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis minus 3")[$y = f ( x ) − 3$] + #math.equation(block: false, alt: "y equals f open parenthesis x minus 2 close parenthesis")[$y = f ( x − 2 )$] + #math.equation(block: false, alt: "y equals f open parenthesis x plus 1 close parenthesis")[$y = f ( x + 1 )$] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Division Algorithm for Polynomials] If #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$] are nonconstant polynomials with real coefficients, then there exist unique polynomials #math.equation(block: false, alt: "q open parenthesis x close parenthesis")[$q ( x )$] and #math.equation(block: false, alt: "r open parenthesis x close parenthesis")[$r ( x )$] such that #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals g open parenthesis x close parenthesis q open parenthesis x close parenthesis plus r open parenthesis x close parenthesis ,")[$f ( x ) = g ( x ) q ( x ) + r ( x ) ,$] where #math.equation(block: false, alt: "deg r open parenthesis x close parenthesis less than deg g open parenthesis x close parenthesis")[$"deg " r ( x ) < "deg " g ( x )$]. ] In Problems 57–60, use polynomial division to divide #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] by #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$], and hence find the quotient, #math.equation(block: false, alt: "q open parenthesis x close parenthesis")[$q ( x )$], and remainder, #math.equation(block: false, alt: "r open parenthesis x close parenthesis")[$r ( x )$]. (See Algebra Skills Refresher Polynomials and Factoring to review polynomial division.) #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 2 x cubed minus 2 x squared minus 19 x minus 11 , g open parenthesis x close parenthesis equals x minus 3")[$f ( x ) = 2 x^(3) − 2 x^(2) − 19 x − 11 , " " g ( x ) = x − 3$] #math.equation(block: true, alt: "q open parenthesis x close parenthesis equals 2 x squared plus 4 x minus 7")[$q ( x ) = 2 x^(2) + 4 x − 7$]; #math.equation(block: true, alt: "r open parenthesis x close parenthesis equals minus 32")[$r ( x ) = − 32$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 3 x cubed plus 12 x squared minus 13 x minus 32 , g open parenthesis x close parenthesis equals x plus 4")[$f ( x ) = 3 x^(3) + 12 x^(2) − 13 x − 32 , " " g ( x ) = x + 4$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x to the power 5 plus 2 x to the power 4 minus 7 x cubed minus 12 x squared plus 5 , g open parenthesis x close parenthesis equals x squared plus 2 x minus 1")[$f ( x ) = x^(5) + 2 x^(4) − 7 x^(3) − 12 x^(2) + 5 , " " g ( x ) = x^(2) + 2 x − 1$] #math.equation(block: true, alt: "q open parenthesis x close parenthesis equals x cubed minus 6 x")[$q ( x ) = x^(3) − 6 x$]; #math.equation(block: true, alt: "r open parenthesis x close parenthesis equals minus 6 x plus 5")[$r ( x ) = − 6 x + 5$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x to the power 5 minus 4 x to the power 4 plus 11 x cubed minus 12 x squared plus 5 x plus 2 , g open parenthesis x close parenthesis equals x squared minus x plus 3")[$f ( x ) = x^(5) − 4 x^(4) + 11 x^(3) − 12 x^(2) + 5 x + 2 , " " g ( x ) = x^(2) − x + 3$] The remainder theorem states: If #math.equation(block: false, alt: "P open parenthesis x close parenthesis")[$P ( x )$] is a polynomial and #math.equation(block: false, alt: "a")[$a$] is any real number, there is a unique polynomial #math.equation(block: false, alt: "Q open parenthesis x close parenthesis")[$Q ( x )$] such that #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals open parenthesis x minus a close parenthesis Q open parenthesis x close parenthesis plus P open parenthesis a close parenthesis")[$P ( x ) = ( x − a ) Q ( x ) + P ( a )$] Follow the steps below to prove the remainder theorem + State the division algorithm applied to the polynomials #math.equation(block: false, alt: "P open parenthesis x close parenthesis")[$P ( x )$] and #math.equation(block: false, alt: "x minus a")[$x − a$]. + What must be the degree of #math.equation(block: false, alt: "r open parenthesis x close parenthesis")[$r ( x )$] in this case? + Evaluate your expression from part (a) at #math.equation(block: false, alt: "x equals a")[$x = a$]. What does this tell you about the remainder, #math.equation(block: false, alt: "r open parenthesis x close parenthesis")[$r ( x )$]? + If #math.equation(block: false, alt: "P open parenthesis x close parenthesis")[$P ( x )$] is a nonconstant polynomial with real coefficients and #math.equation(block: false, alt: "a")[$a$] is any real number, then there exist unique polynomials #math.equation(block: false, alt: "q open parenthesis x close parenthesis")[$q ( x )$] and #math.equation(block: false, alt: "r open parenthesis x close parenthesis")[$r ( x )$] such that #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals open parenthesis x minus a close parenthesis q open parenthesis x close parenthesis plus r open parenthesis x close parenthesis")[$P ( x ) = ( x − a ) q ( x ) + r ( x )$] where #math.equation(block: false, alt: "deg r open parenthesis x close parenthesis less than deg open parenthesis x minus a close parenthesis")[$"deg " r ( x ) < "deg " ( x − a )$]. + Zero + #math.equation(block: false, alt: "P open parenthesis a close parenthesis equals open parenthesis a minus a close parenthesis q open parenthesis a close parenthesis plus r open parenthesis a close parenthesis equals r open parenthesis a close parenthesis")[$P ( a ) = ( a − a ) q ( a ) + r ( a ) = r ( a )$]. Because #math.equation(block: false, alt: "deg r open parenthesis x close parenthesis equals 0")[$"deg " r ( x ) = 0$], #math.equation(block: false, alt: "r open parenthesis x close parenthesis")[$r ( x )$] is a constant. That constant value is #math.equation(block: false, alt: "P open parenthesis a close parenthesis")[$P ( a )$], so #math.equation(block: false, alt: "P open parenthesis x close parenthesis equals open parenthesis x minus a close parenthesis q open parenthesis x close parenthesis plus P open parenthesis a close parenthesis")[$P ( x ) = ( x − a ) q ( x ) + P ( a )$]. Verify the remainder theorem for the following: + #math.equation(block: false, alt: "P open parenthesis x close parenthesis equals x cubed minus 4 x squared plus 2 x minus 1 , a equals 2")[$P ( x ) = x^(3) − 4 x^(2) + 2 x − 1 , a = 2$] + #math.equation(block: false, alt: "P open parenthesis x close parenthesis equals 3 x squared plus x minus 5 , a equals minus 3")[$P ( x ) = 3 x^(2) + x − 5 , " " a = − 3$] Use the remainder theorem to prove the factor theorem, stated earlier in this section. You will need to justify two statements: + If #math.equation(block: false, alt: "P open parenthesis a close parenthesis equals 0")[$P ( a ) = 0$], show that #math.equation(block: false, alt: "x minus a")[$x − a$] is a factor of #math.equation(block: false, alt: "P open parenthesis x close parenthesis")[$P ( x )$]. + If #math.equation(block: false, alt: "x minus a")[$x − a$] is a factor of #math.equation(block: false, alt: "P open parenthesis x close parenthesis")[$P ( x )$], show that #math.equation(block: false, alt: "P open parenthesis a close parenthesis equals 0")[$P ( a ) = 0$]. + From the remainder theorem, #math.equation(block: false, alt: "P open parenthesis x close parenthesis, equals open parenthesis x minus a close parenthesis Q open parenthesis x close parenthesis plus P open parenthesis a close parenthesis; equals open parenthesis x minus a close parenthesis Q open parenthesis x close parenthesis plus 0; equals open parenthesis x minus a close parenthesis Q open parenthesis x close parenthesis")[$P ( x ) & = ( x − a ) Q ( x ) + P ( a ) \ & = ( x − a ) Q ( x ) + 0 \ & = ( x − a ) Q ( x )$] + By definition of a factor, if #math.equation(block: false, alt: "x minus a")[$x − a$] is a factor of #math.equation(block: false, alt: "P open parenthesis x close parenthesis")[$P ( x )$], then #math.equation(block: false, alt: "P open parenthesis x close parenthesis equals open parenthesis x minus a close parenthesis q open parenthesis x close parenthesis")[$P ( x ) = ( x − a ) q ( x )$], so #math.equation(block: false, alt: "P open parenthesis x close parenthesis equals open parenthesis x minus a close parenthesis q open parenthesis x close parenthesis plus 0")[$P ( x ) = ( x − a ) q ( x ) + 0$]. The uniqueness guaranteed in the remainder theorem tells us that #math.equation(block: false, alt: "P open parenthesis a close parenthesis equals 0")[$P ( a ) = 0$]. Verify the factor theorem for the following: + #math.equation(block: false, alt: "P open parenthesis x close parenthesis equals x to the power 4 minus 4 x cubed minus 11 x squared plus 3 x plus 2 , a equals minus 2")[$P ( x ) = x^(4) − 4 x^(3) − 11 x^(2) + 3 x + 2 , " " a = − 2$] + #math.equation(block: false, alt: "P open parenthesis x close parenthesis equals x cubed plus 2 x squared minus 31 x minus 20 , a equals 5")[$P ( x ) = x^(3) + 2 x^(2) − 31 x − 20 , " " a = 5$] For Problems 65–68, + Verify that the given value is a zero of the polynomial. + Find the other zeros. (#emph[Hint]: Use polynomial division to write #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals open parenthesis x minus a close parenthesis Q open parenthesis x close parenthesis")[$P ( x ) = ( x − a ) Q ( x )$] then factor #math.equation(block: false, alt: "Q open parenthesis x close parenthesis .")[$Q ( x ) .$]) #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals x cubed minus 2 x squared plus 1")[$P ( x ) = x^(3) − 2 x^(2) + 1$]; #math.equation(block: true, alt: "a equals 1")[$a = 1$] + #math.equation(block: false, alt: "P open parenthesis 1 close parenthesis equals 0")[$P ( 1 ) = 0$] + #math.equation(block: false, alt: "the fraction 1 plus or minus the square root of 5 over 2")[$display(frac(1 ± sqrt(5), 2))$] #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals x cubed plus 2 x squared minus 1")[$P ( x ) = x^(3) + 2 x^(2) − 1$]; #math.equation(block: true, alt: "a equals minus 1")[$a = − 1$] #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals x to the power 4 minus 3 x cubed minus 10 x squared plus 24 x")[$P ( x ) = x^(4) − 3 x^(3) − 10 x^(2) + 24 x$]; #math.equation(block: true, alt: "a equals minus 3")[$a = − 3$] + #math.equation(block: false, alt: "P open parenthesis minus 3 close parenthesis equals 0")[$P ( − 3 ) = 0$] + #math.equation(block: false, alt: "0 , 2 , 4")[$0 , 2 , 4$] #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals x to the power 4 plus 5 x cubed minus x squared minus 5 x")[$P ( x ) = x^(4) + 5 x^(3) − x^(2) − 5 x$]; #math.equation(block: true, alt: "a equals minus 5")[$a = − 5$] In Problems 69–70, we use polynomials to approximate other functions. + Graph the functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals e to the power x")[$f ( x ) = e^(x)$] and #math.equation(block: true, alt: "p open parenthesis x close parenthesis equals 1 plus x plus the fraction 1 over 2 x squared plus the fraction 1 over 6 x cubed")[$p ( x ) = 1 + x + frac(1, 2) x^(2) + frac(1, 6) x^(3)$] in the standard window. For what values of #math.equation(block: false, alt: "x")[$x$] does it appear that #math.equation(block: false, alt: "p open parenthesis x close parenthesis")[$p ( x )$] would be a good approximation for #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]? + Change the window settings to #math.equation(block: true, alt: "Xmin, equals minus 4.7, Xmax equals 4.7; Ymin, equals 0, Ymax equals 20")[$"Xmin" & = − 4.7 & & "Xmax" = 4.7 \ "Ymin" & = 0 & & "Ymax" = 20$] and fill in the table of values below. (You can use the #strong[value] feature on your calculator.)#figure(table( columns: 8, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus 0.5")[$− 0.5$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0.5")[$0.5$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "1.5")[$1.5$]], [#math.equation(block: false, alt: "2")[$2$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [#math.equation(block: false, alt: "p open parenthesis x close parenthesis")[$p ( x )$]], [], [], [], [], [], [], [], )) + The #strong[error] in the approximation is the difference #math.equation(block: false, alt: "f open parenthesis x close parenthesis minus p open parenthesis x close parenthesis")[$f ( x ) − p ( x )$]. We can reduce the error by using a polynomial of higher degree. The #math.equation(block: false, alt: "n")[$n$]th degree polynomial for approximating #math.equation(block: false, alt: "e to the power x")[$e^(x)$] is #math.equation(block: true, alt: "P sub n open parenthesis x close parenthesis equals 1 plus x plus the fraction 1 over 2 ! x squared plus the fraction 1 over 3 ! x cubed plus ⋯ plus the fraction 1 over n ! x to the power n")[$P_(n) ( x ) = 1 + x + frac(1, 2 !) x^(2) + frac(1, 3 !) x^(3) + ⋯ + frac(1, n !) x^(n)$] where #math.equation(block: false, alt: "n ! equals n open parenthesis n minus 1 close parenthesis open parenthesis n minus 2 close parenthesis ⋯ 3 times 2 times 1")[$n ! = n ( n − 1 ) ( n − 2 ) ⋯ 3 ⋅ 2 ⋅ 1$]. Graph #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] and #math.equation(block: false, alt: "P sub 5 open parenthesis x close parenthesis")[$P_(5) ( x )$] in the same window as in part (b). What is the error in approximating #math.equation(block: false, alt: "f open parenthesis 2 close parenthesis")[$f ( 2 )$] by #math.equation(block: false, alt: "P sub 5 open parenthesis 2 close parenthesis")[$P_(5) ( 2 )$]? + Graph #math.equation(block: false, alt: "f open parenthesis x close parenthesis minus P sub 5 open parenthesis x close parenthesis")[$f ( x ) − P_(5) ( x )$] in the same window as in part (b). What does the graph tell you about the error in approximating #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] by #math.equation(block: false, alt: "P sub 5 open parenthesis x close parenthesis")[$P_(5) ( x )$]? + About #math.equation(block: false, alt: "minus 1 less than x less than 2")[$− 1 < x < 2$]#figure(figph[cubic on exponential], alt: "cubic on exponential", caption: none) + #figure(table( columns: 8, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus 0.5")[$− 0.5$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0.5")[$0.5$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "1.5")[$1.5$]], [#math.equation(block: false, alt: "2")[$2$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [#math.equation(block: false, alt: "0.368")[$0.368$]], [#math.equation(block: false, alt: "0.607")[$0.607$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "1.649")[$1.649$]], [#math.equation(block: false, alt: "2.718")[$2.718$]], [#math.equation(block: false, alt: "4.482")[$4.482$]], [#math.equation(block: false, alt: "7.389")[$7.389$]], [#math.equation(block: false, alt: "p open parenthesis x close parenthesis")[$p ( x )$]], [#math.equation(block: false, alt: "0.333")[$0.333$]], [#math.equation(block: false, alt: "0.604")[$0.604$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "1.646")[$1.646$]], [#math.equation(block: false, alt: "2.667")[$2.667$]], [#math.equation(block: false, alt: "4.188")[$4.188$]], [#math.equation(block: false, alt: "6.333")[$6.333$]], )) + #math.equation(block: false, alt: "0.122")[$0.122$] + #figure(figph[quintic on exponential], alt: "quintic on exponential", caption: none) #linebreak() The error is relatively small for values of #math.equation(block: false, alt: "x")[$x$] between #math.equation(block: false, alt: "minus 3")[$− 3$] and #math.equation(block: false, alt: "2.5")[$2.5$]. In Projects for Chapter 2: Periodic Functions, we investigated periodic functions. The #strong[sine function], #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals sin ⁡ open parenthesis x close parenthesis")[$f ( x ) = sin ( x )$], is a useful periodic function. + Graph the functions #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals sin ⁡ open parenthesis x close parenthesis and p open parenthesis x close parenthesis equals x minus the fraction 1 over 6 x cubed")[$f ( x ) = sin ( x ) " " " " " " " " "and" " " " " " " " " p ( x ) = x − frac(1, 6) x^(3)$] in the standard window. (Check that your calculator is set in #strong[Radian] mode.) For what values of #math.equation(block: false, alt: "x")[$x$] does it appear that #math.equation(block: false, alt: "p open parenthesis x close parenthesis")[$p ( x )$] would be a good approximation for #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]? + Change the window settings to #math.equation(block: true, alt: "Xmin, equals minus 4.7, Xmax equals 4.7; Ymin, equals minus 2, Ymax equals 2")[$"Xmin" & = − 4.7 & & "Xmax" = 4.7 \ "Ymin" & = − 2 & & "Ymax" = 2$] and fill in the table of values below. (You can use the #strong[value] feature on your calculator.)#figure(table( columns: 8, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus 0.5")[$− 0.5$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0.5")[$0.5$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "1.5")[$1.5$]], [#math.equation(block: false, alt: "2")[$2$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [#math.equation(block: false, alt: "p open parenthesis x close parenthesis")[$p ( x )$]], [], [], [], [], [], [], [], )) + Two more polynomials for approximating #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals sin ⁡ open parenthesis x close parenthesis")[$f ( x ) = sin ( x )$] are #math.equation(block: true, alt: "P sub 5 open parenthesis x close parenthesis, equals 1 minus the fraction 1 over 3 ! x cubed plus the fraction 1 over 5 ! x to the power 5; P sub 7 open parenthesis x close parenthesis, equals 1 minus the fraction 1 over 3 ! x cubed plus the fraction 1 over 5 ! x to the power 5 minus plus the fraction 1 over 7 ! x to the power 7")[$P_(5) ( x ) & = 1 − frac(1, 3 !) x^(3) + frac(1, 5 !) x^(5) \ P_(7) ( x ) & = 1 − frac(1, 3 !) x^(3) + frac(1, 5 !) x^(5) − + frac(1, 7 !) x^(7)$] (See Problem for the definition of #math.equation(block: false, alt: "n !")[$n !$].) Graph #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] and #math.equation(block: false, alt: "P sub 5 open parenthesis x close parenthesis")[$P_(5) ( x )$] in the same window as in part (b). What is the error in approximating #math.equation(block: false, alt: "f open parenthesis 2 close parenthesis")[$f ( 2 )$] by #math.equation(block: false, alt: "P sub 5 open parenthesis 2 close parenthesis")[$P_(5) ( 2 )$]? + Graph #math.equation(block: false, alt: "f open parenthesis x close parenthesis minus P sub 5 open parenthesis x close parenthesis")[$f ( x ) − P_(5) ( x )$] in the same window as in part (b). What does the graph tell you about the error in approximating #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] by #math.equation(block: false, alt: "P sub 5 open parenthesis x close parenthesis")[$P_(5) ( x )$]?