6.4 Problem Solving

Many quadratic models arise as the product of two variables, one of which increases while the other decreases. For example, the area of a rectangle is the product of its length and its width, or $A = l w$ . If we require that the rectangle have a certain perimeter, then as we increase its length, we must also decrease its width. (We analyzed this problem in Perimeter and Area of Modeling with Functions.)

For Revenue from Theater Tickets, recall the formula for the revenue from sales of an item:

$Revenue = (price of one item) (number of items sold)$

Usually, when the price of an item increases, the number of items sold decreases.

Investigation

Revenue from Theater Tickets

The local theater group sold tickets to its opening night performance for $$ 5$ and drew an audience of $100$ people. The next night, the group reduced the ticket price by $$ 0.25$ and $10$ more people attended; that is, $110$ people bought tickets at $$ 4.75$ apiece. In fact, for each $$ 0.25$ reduction in ticket price, $10$ additional tickets can be sold.

Complete the table.
Number
of price
reductions
Price of
ticket
Number
of tickets
sold
Total
revenue
$0$ $5.00$ $100$ $500$
$1$ $4.75$ $110$ $522.50$
$2$
$3$
$4$
$5$
$6$
$7$
$8$
$9$
$10$
$11$
$12$
Use your table to make a graph. Plot total revenue on the vertical axis versus number of price reductions on the horizontal axis.
Let $x$ represent the number of price reductions, as in the first column of the table. Write algebraic expressions in terms of $x$ for each quantity.
The price of a ticket after $x$ price reductions:
$Price =$

The number of tickets sold at that price:
$Number =$

The total revenue from ticket sales:
$Revenue =$
Enter your expressions for the price of a ticket, the number of tickets sold, and the total revenue into the calculator as $Y sub 1$ , $Y sub 2$ , and $Y sub 3$ . Use the Table feature to verify that your algebraic expressions agree with your table from part (1).
Use your calculator to graph your expression for total revenue in terms of $x$ . Use your table to choose appropriate window settings that show the high point of the graph and both xintercepts.
What is the maximum revenue possible from ticket sales? What price should the theater group charge for a ticket to generate that revenue? How many tickets will the group sell at that price?

Maximum or Minimum Values

Finding the maximum or minimum value for a variable expression is a common problem in many applications. For example, if you own a company that manufactures blue jeans, you might like to know how much to charge for your jeans in order to maximize your revenue.

As you increase the price of the jeans, your revenue may increase for a while. But if you charge too much for the jeans, consumers will not buy as many pairs, and your revenue may actually start to decrease. Is there some optimum price you should charge for a pair of jeans in order to achieve the greatest revenue?

Example 1

Late Nite Blues finds that it can sell $600 minus 15 x$ pairs of jeans per week if it charges $x$ dollars per pair. (Notice that as the price increases, the number of pairs of jeans sold decreases.)

Write an equation for the revenue as a function of the price of a pair of jeans.
Graph the function.
How much should Late Nite Blues charge for a pair of jeans in order to maximize its revenue?

Using the formula for revenue stated above, we find
$\begin{aligned} Revenue & = (price of one item) (number of items sold) \\ R & = x (600 - 15 x) \\ R & = 600 x - 15 x^{2} \end{aligned}$
We recognize the function as quadratic, so the graph is a parabola. You can use your calculator to verify the graph below.
The maximum value of $R$ occurs at the vertex of the parabola. Thus,
$\begin{aligned} x_{v} & = \frac{- b}{2 a} = \frac{- 600}{2 (- 15)} = 20 \\ y_{v} & = 600 (20) - 15 (20)^{2} = 6000 \end{aligned}$
The revenue takes on its maximum value when $x equals 20$ , and the maximum value is $R equals 6000$ . This means that Late Nite Blues should charge $ $20$ for a pair of jeans in order to maximize revenue at $ $6000$ a week.

The height of golf ball in meters is given by $h = - 4.9 t^{2} + 20 t$ . What does the vertex of the graph tell us about the golf ball?

_____

When the golf ball reaches its maximum height

The height of golf ball in meters is given by $h = - 4.9 t^{2} + 20 t$ . What does the vertex of the graph tell us about the golf ball?

How long the golf ball is in the air
The starting height of the golf ball
When the golf ball reaches its maximum height
The speed of the golf ball

The Metro Rail service sells $1200 minus 80 x$ tickets each day when it charges $x$ dollars per ticket.

Write an equation for the revenue, $R$ , as a function of the price of a ticket.
$R =$ _____
What ticket price will return the maximum revenue?
$_____
What is the maximum revenue?
$_____

The revenue is the product of the number of tickets times the price per ticket: $R = 1200 x - 80 x^{2}$
$ $7.50,$ $ $4500$

The Metro Rail service sells $1200 minus 80 x$ tickets each day when it charges $x$ dollars per ticket.

Write an equation for the revenue, $R$ , as a function of the price of a ticket.
What ticket price will return the maximum revenue? What is the maximum revenue?

The revenue is the product of the number of tickets times the price per ticket: $R = 1200 x - 80 x^{2}$
ticket price: $ $7.50$ , maximum revenue: $ $4500$

Explain why revenue will probably not increase indefinitely as price increases.

_____

Explain why revenue will probably not increase indefinitely as price increases.

The Vertex Form for a Parabola

What is the $x$ -coordinate of the vertex of the parabola $y = 4 (x - 2) (x + 8)$ ?

_____

$minus 3$

What is the $x$ -coordinate of the vertex of the parabola $y = 4 (x - 2) (x + 8)$ ?

$4$
$minus 16$
$6$
$minus 3$

Consider the quadratic equation

$y = 2 (x - 3)^{2} - 8$

By expanding the squared expression and collecting like terms, we can rewrite the equation in standard form as

$\begin{aligned} y & = 2 (x^{2} - 6 x + 9) - 8 \\ y & = 2 x^{2} - 12 x + 10 \end{aligned}$

The vertex of this parabola is

$\begin{aligned} x_{v} & = \frac{- (- 12)}{2 (2)} = 3 \\ y_{v} & = 2 (3)^{2} - 12 (3) + 10 = - 8 \end{aligned}$

and its graph is shown below.

Notice that the coordinates of the vertex, $open parenthesis 3, minus 8 close parenthesis$ , are apparent in the original equation; we don’t need to do any computation to find the vertex.

$\begin{aligned} y = 2 (x - & 3)^{2} & - 8 \\ x_{v} & y_{v} \end{aligned}$

This equation is an example of the vertex form for a quadratic function.

To understand why the vertex form works, substitute $x sub v equals 4$ into $y = - 3 (x - 4)^{2} + 6$ from Example to find

$y = - 3 (4 - 4)^{2} + 6 = 6$

which confirms that when $x equals 4$ , $y equals 6$ . Next, notice that if $x$ is any number except $4$ , the expression $- 3 (x - 4)^{2}$ is negative, so $y less than 6$ . Therefore, $6$ is the maximum value for $y$ on the graph, so $open parenthesis 4, 6 close parenthesis$ is the high point or vertex.

You can also rewrite $y = - 3 (x - 4)^{2} + 6$ in standard form and use the formula $x_{v} = \frac{- b}{2 a}$ to confirm that the vertex is the point $open parenthesis 4, 6 close parenthesis$ .

Find the vertex of the graph of $y = 5 - \frac{1}{2} (x + 2)^{2}$ .
Vertex: _____
Write the equation of the parabola in standard form.
$y =$ _____

$open parenthesis minus 2, 5 close parenthesis$
$y = - \frac{1}{2} x^{2} - 2 x + 3$

Find the vertex of the graph of $y = 5 - \frac{1}{2} (x + 2)^{2}$ .
Write the equation of the parabola in standard form.

$open parenthesis minus 2, 5 close parenthesis$
$y = - \frac{1}{2} x^{2} - 2 x + 3$

What is the smallest $y$ -value on the graph of $y = 4 (x - 6)^{2} + 12$ ?

_____

$12$

What is the smallest $y$ -value on the graph of $y = 4 (x - 6)^{2} + 12$ ?

$6$
$4$
$12$
We can't tell without graphing

Any quadratic equation in vertex form can be written in standard form by expanding, and any quadratic equation in standard form can be put into vertex form by completing the square.

Example 3

Write the equation $y = 3 x^{2} - 6 x - 1$ in vertex form and find the vertex of its graph.

We factor the lead coefficient, $3$ , from the variable terms, leaving a space to complete the square.

$y = 3 (x^{2} - 2 x + \underset{―}{}) - 1$

Next, we complete the square inside parentheses. Take half the coefficient of $x$ and square the result:

$p = \frac{1}{2} (- 2) = - 1, and p^{2} = (- 1)^{2} = 1$

. We must add $1$ to complete the square. However, we are really adding $3 open parenthesis 1 close parenthesis$ to the right side of the equation, so we must also subtract $3$ to compensate:

$y = 3 (x^{2} - 2 x + 1) - 1 - 3$

The expression inside parentheses is now a perfect square, and the vertex form is

$y = 3 (x - 1)^{2} - 4$

The vertex of the parabola is $open parenthesis 1, minus 4 close parenthesis$ .

Write the equation $y = 2 x^{2} + 12 x + 13$ in vertex form, and find the vertex of its graph.

The vertex form is $y = a (x - x_{v})^{2} + y_{v}$ , where

$a =$ _____

$x_{v} =$ _____

$y_{v} =$ _____

The vertex of the graph is _____

Factor $2$ from the variable terms.
Complete the square inside parentheses.
Subtract $2 p squared$ outside parentheses.
Write the vertex form.

$y = 2 (x + 3)^{2} - 5$ ; $(- 3, - 5)$

Write the equation $y = 2 x^{2} + 12 x + 13$ in vertex form, and find the vertex of its graph.

Factor $2$ from the variable terms.
Complete the square inside parentheses.
Subtract $2 p squared$ outside parentheses.
Write the vertex form.

$y = 2 (x + 3)^{2} - 5$ ; vertex: $(- 3, - 5)$

Why do we need to know a second point besides the vertex to find the equation of a parabola?

_____

Why do we need to know a second point besides the vertex to find the equation of a parabola?

Graphing with the Vertex Form

We can also use the vertex form to sketch a graph, using what we know about transformations.

Example 4

Use transformations to graph $f (x) = - 3 (x - 4)^{2} + 6$ .

We can graph $f (x)$ by applying transformations to the basic parabola, $y = x^{2}$ . To identify the transformations, think of evaluating the function for a specific value of $x$ . What operations would we perform on $x$ , besides squaring?

1. Subtract 4:	$y = (x - 4)^{2}$	$Shift 4 units right.$
2. Multiply by -3:	$y = - 3 (x - 4)^{2}$	$Stretch by a factor of 3,$ $and reflect about the x -axis.$
3. Add 6:	$y = - 3 (x - 4)^{2} + 6$	$Shift up 6 units.$

We perform the same transformations on the graph of $y = x^{2}$ , as shown in the figure.

List the transformations of y = x 2 needed to graph g ( x ) = 5 − 1 2 ( x + 2 ) 2 .
- Shift 2 units _____
- Reflect about the $x$ -axis and _____ by a factor of $\frac{1}{2}$ .
- Shift 5 units _____
Use transformations to sketch the graph.

Shift 2 units left, reflect about $x$ -axis and compress by a factor of 2, shift 5 units up.
A graph is below.

List the transformations of $y = x^{2}$ needed to graph $g (x) = 5 - \frac{1}{2} (x + 2)^{2}$ .
Use transformations to sketch the graph.

Shift 2 units left, reflect about $x$ -axis and compress by a factor of 2, shift 5 units up.

Systems Involving Quadratic Equations

Recall that the solution to a $2 times 2$ system of linear equations is the intersection point of the graphs of the equations. (See Algebra Review Refresher Linear Systems in Two Variables.) This is also true of systems in which one or both of the equations is quadratic. The figure below shows the three cases for systems of one quadratic and one linear equation.

In Example, we use both graphical and algebraic techniques to solve the system.

Example 5

The Pizza Connection calculates that the cost, in dollars, of producing $x$ pizzas per day is given by

$C = 0.15 x^{2} + 0.75 x + 180$

The Pizza Connection charges $ $15$ per pizza, so the revenue from selling $x$ pizzas is

$R equals 15 x$

How many pizzas per day must the Pizza Connection sell in order to break even?

To break even means to make zero profit. Because

$Profit = Revenue - Cost$

the break-even points occur when revenue equals cost. In mathematical terms, we would like to find any values of $x$ for which $R = C$ .

If we graph the revenue and cost functions on the same axes, these values correspond to points where the two graphs intersect. Use the window settings

$\begin{aligned} Xmin & = 0 & Xmax = 94 \\ Ymin & = 0 & Ymax = 1400 \end{aligned}$

on your calculator to obtain the graph shown below. You can verify that the two intersection points are $open parenthesis 15, 225 close parenthesis$ and $open parenthesis 80, 1200 close parenthesis$ .

Thus, the Pizza Connection must sell either $15$ or $80$ pizzas in order to break even. On the graph we see that revenue is greater than cost for $x$ -values between $15$ and $80$ , so the Pizza Connection will make a profit if it sells between $15$ and $80$ pizzas.

We can also solve algebraically for the break-even points. The intersection points of the two graphs correspond to the solutions of the system of equations

$\begin{aligned} y & = 0.15 x^{2} + 0.75 x + 180 \\ y & = 15 x \end{aligned}$

We equate the two expressions for $y$ and solve for $x$ :

$\begin{aligned} 0.15 x^{2} + 0.75 x + 180 & = 15 x & Subtract 15 x from both sides. \\ 0.15 x^{2} - 14.25 x + 180 & = 0 & Use the quadratic formula. \end{aligned}$

$\begin{aligned} x & = \frac{14.25 \pm \sqrt{{14.25}^{2} - 4 (0.15) (180)}}{2 (- 0.05)} & Simplify. \\ = \frac{14.25 \pm 9.75}{0.3} \end{aligned}$

The solutions are $15$ and $80$ , as we found from the graph.

Solve the system algebraically:
$\begin{aligned} y & = x^{2} - 6 x - 7 \\ y & = 13 - x^{2} \end{aligned}$

Solutions: _____ Note: list solutions as ordered pairs, and use a comma to separate solutions.
Graph both equations, and show the solutions on the graph.

$(- 2, 9), (5, - 12)$

Solve the system algebraically:
$\begin{aligned} y & = x^{2} - 6 x - 7 \\ y & = 13 - x^{2} \end{aligned}$
Graph both equations, and show the solutions on the graph.

$(- 2, 9), (5, - 12)$

Explain how to find the intersection points of two parabolas.

_____

Explain how to find the intersection points of two parabolas.

Technology

Solving Systems with the Graphing Calculator

We can use the intersect feature of the graphing calculator to solve systems of quadratic equations. Consider the system

$\begin{aligned} y & = (x + 1.1)^{2} \\ y & = 7.825 - 2 x - 2.5 x^{2} \end{aligned}$

We will graph these two equations in the standard window. The two intersection points are visible in the window, but we do not find their exact coordinates when we trace the graphs.

We can use the intersect command to locate one of the solutions, as shown below. You can check that the point $open parenthesis 0.9, 4 close parenthesis$ is an exact solution to the system by substituting $x equals 0.9$ and $y equals 4$ into each equation of the system. (The calculator is not always able to find the exact coordinates, but it usually gives a very good approximation.)

You can find the other solution of the system by following the same steps and moving the bug close to the other intersection point. You should verify that the other solution is the point $open parenthesis minus 2.1, 1 close parenthesis$ .

Section Summary

Vocabulary

Look up the definitions of new terms in the Glossary.

Maximum value
Minimum value
Vertex form

CONCEPTS

Quadratic models may arise as the product of two variables.
The maximum or minimum of a quadratic function occurs at the vertex.
Note
Vertex Form for a Quadratic Function
A quadratic function $y = a x^{2} + b x + c$ , $a not equal to 0$ , can be written in the vertex form
$y = a (x - x_{v})^{2} + y_{v}$
where the vertex of the graph is $(x_{v}, y_{v})$ .
We can convert a quadratic equation to vertex form by completing the square.
We can graph a quadratic equation in vertex form using transformations.
A $2 times 2$ system involving quadratic equations may have one, two, or no solutions.

STUDY QUESTIONS

How can you tell whether a variable given by a quadratic equation has a maximum value or a minimum value?
Correct the following false statement.
- The maximum or minimum value given by a quadratic equation is the average of the $x$ -intercepts. ( $False!$ )
Explain why $minus 4$ is the smallest function value for $f (x) = 2 (x - 3)^{2} - 4$ .
In the equation $y = \frac{1}{3} (x + 5)^{2} - 2$ , what does each of the constants tell you about the graph?
Francine attempts to write the equation $g (x) = 2 x^{2} - 6 x + 1$ in vertex form as follows: $g (x) = (2 x^{2} - 6 x + 9) + 1 - 9$ . What is wrong with her work?
Without doing any calculations, solve the system $y = x^{2} + 4, y = 2 - 3 x^{2}$ . (Hint: Visualize the graphs.)

SKILLS

Practice each skill in the Homework problems listed.

Find the maximum or minimum value of a quadratic function: #1–14
Convert a quadratic equation from vertex form to standard form: #19–22
Convert a quadratic equation from standard form to vertex form: #23–28
Use transformations to graph a quadratic equation: #15–28
USolve a system involving quadratic equations: #31–50

Homework 6.4

The owner of a motel has $60$ rooms to rent. She finds that if she charges $ $0$ per room per night, all the rooms will be rented. For every $ $2$ that she increases the price of a room, $3$ rooms will stand vacant.

Complete the table. The first two rows are filled in for you.
No. of price
increases
Price of
room
No. of rooms
rented
Total
revenue
$0$ $20$ $60$ $1200$
$1$ $22$ $57$ $1254$
$2$
$3$
$4$
$5$
$6$
$7$
$8$
$10$
$12$
$16$
$20$
Let $x$ stand for the number of $$ 2$ price increases the owner makes. Write algebraic expressions for the price of a room, the number of rooms that will be rented, and the total revenue earned at that price.
Use your calculator to make a table of values for your algebraic expressions. Let $Y sub 1$ stand for the price of a room, $Y sub 2$ for the number of rooms rented, and $Y sub 3$ for the total revenue. Verify the values you calculated in part (a).
Use your table to find a value of $x$ that causes the total revenue to be zero.
Use your graphing calculator to graph your formula for total revenue.
What is the lowest price that the owner can charge for a room if she wants her revenue to exceed $$ 1296$ per night? What is the highest price she can charge to obtain this revenue?
What is the maximum revenue the owner can earn in one night? How much should she charge for a room to maximize her revenue? How many rooms will she rent at that price?

No. of price increases	Price of room	No. of rooms rented	Total revenue
$0$	$20$	$60$	$1200$
$1$	$22$	$57$	$1254$
$2$	$24$	$54$	$1296$
$3$	$26$	$51$	$1326$
$4$	$28$	$48$	$1344$
$5$	$30$	$45$	$1350$
$6$	$32$	$42$	$1344$
$7$	$34$	$39$	$1326$
$8$	$36$	$36$	$1296$
$10$	$40$	$30$	$1200$
$12$	$44$	$24$	$1056$
$16$	$52$	$12$	$624$
$20$	$60$	$0$	$0$

Price of a room: $20 plus 2 x$ ; Rooms rented: $60 minus 3 x$ ; Revenue: $1200 + 60 x - 6 x^{2}$
$20$
$ $24;$ $ $36$
$ $1350;$ $ $30;$ $45$ rooms

The owner of a video store sells $96$ blank tapes per week if he charges $$ 6$ per tape. For every $$ 0.50$ he increases the price, he sells $4$ fewer tapes per week.

Complete the table. The first two rows are filled in for you.
No. of price
increases
Price of
tape
No. of tapes
sold
Total
revenue
$0$ $6$ $96$ $576$
$1$ $6.50$ $92$ $598$
$2$
$3$
$4$
$5$
$6$
$7$
$8$
$12$
$16$
$20$
$24$
Let $x$ stand for the number of $$ 0.50$ price increases the owner makes. Write algebraic expressions for the price of a tape, the number of tapes sold, and the total revenue.
Use your calculator to make a table of values for your algebraic expressions. Let $Y sub 1$ stand for the price of a tape, $Y sub 2$ for the number of tapes sold, and $Y sub 3$ for the total revenue. Verify the values you calculated in part (a).
Use your table to find a value of $x$ that causes the total revenue to be zero.
Use your graphing calculator to graph your formula for total revenue.
How much should the owner charge for a tape in order to bring in $$ 630$ per week from tapes? (You should have two answers.)
What is the maximum revenue the owner can earn from tapes in one week? How much should he charge for a tape to maximize his revenue? How many tapes will he sell at that price?

Give the dimensions of two different rectangles with perimeter $60$ meters. Compute the areas of the two rectangles.
A rectangle has a perimeter of $60$ meters. If the length of the rectangle is $x$ meters, write an expression for its width.
Write an expression for the area of the rectangle.

(For example) $10$ m by $20$ m with area $200$ sq m; or $15$ m by $15$ m, area $225$ sq m
$30 minus x$
$30 x minus x squared$

Give the dimensions of two different rectangles with perimeter $48$ inches. Compute the areas of the two rectangles.
A rectangle has a perimeter of $48$ inches. If the width of the rectangle is $w$ inches, write an expression for its length.
Write an expression for the area of the rectangle.

For Problems 5–8,

Find the maximum or minimum value algebraically.
Obtain a good graph on your calculator and verify your answer. (Use the coordinates of the vertex and the vertical intercept to help you choose an appropriate window for the graph.)

Delbert launches a toy water rocket from ground level. Its distance above the ground $t$ seconds after launch is given, in feet, by

$d = 96 t - 16 t^{2}$

When will the rocket reach its greatest height, and what will that height be?

$3$ sec, $144$ ft

Francine throws a wrench into the air from the bottom of a trench $12$ feet deep. Its height $t$ seconds later is given, in feet, by

$h = - 12 + 32 t - 16 t^{2}$

When will the wrench reach its greatest height, and what will that height be?

The owners of a small fruit orchard decide to produce gift baskets as a sideline. The cost per basket for producing $x$ baskets is

$C = 0.01 x^{2} - 2 x + 120$

How many baskets should they produce in order to minimize the cost per basket? What will their total cost be at that production level?

$100$ baskets, $$ 2000$

A new electronics firm is considering marketing a line of telephones. The cost per phone for producing $x$ telephones is

$C = 0.001 x^{2} - 3 x + 2270$

How many telephones should the firm produce in order to minimize the cost per phone? What will the firm's total cost be at that production level?

As part of a collage for her art class, Sheila wants to enclose a rectangle with $100$ inches of yarn.

Let $w$ represent the width of the rectangle, and write an expression for its length. Then write an expression that gives the area, $A$ , of the rectangle as a function of its width, $w$ .
What is the area of the largest rectangle that Sheila can enclose with $100$ inches of yarn?

Length: $50 minus w$ ; Area: $50 w minus w squared$
$625$ sq in

Gavin has rented space for a booth at the county fair. As part of his display, he wants to rope off a rectangular area with $80$ yards of rope.

Let $w$ represent the width of the roped-off rectangle, and write an expression for its length. Then write an expression that gives the area, $A$ , of the roped-off space as a function of its width, $w$ .
What is the largest area that Gavin can rope off? What will the dimensions of the rectangle be?

A farmer plans to fence a rectangular grazing area along a river with 300 yards of fence as shown in the figure.

Write an expression that gives the area, $A$ , of the grazing land as a function of the width, $w$ , of the rectangle.
What is the largest area the farmer can enclose?

$300 w minus 2 w squared$
$11, 250$ sq yd

A breeder of horses wants to fence two rectangular grazing areas along a river with $600$ meters of fence as shown in the figure.

Write an expression that gives the area, $A$ , of the grazing land as a function of the width, $w$ , of the rectangles.
What is the largest area the breeder can enclose?

A travel agent offers a group rate of $$ 2400$ per person for a week in London if $16$ people sign up for the tour. For each additional person who signs up, the price per person is reduced by $$ 100$ .

Let $x$ represent the number of additional people who sign up. Write expressions for the total number of people signed up, the price per person, and the total revenue.
How many people must sign up for the tour in order for the travel agent to maximize her revenue?

Number of people: $16 plus x$ ; Price per person: $2400 minus 100 x$ ; Total revenue: $38, 400 + 800 x - 100 x^{2}$
$20$

An entrepreneur buys an apartment building with $40$ units. The previous owner charged $$ 240$ per month for a single apartment and on the average rented $32$ apartments at that price. The entrepreneur discovers that for every $$ 20$ he raises the price, another apartment stands vacant.

Let $x$ represent the number of $$ 20$ price increases. Write expressions for the new price, the number of rented apartments, and the total revenue.
What price should the entrepreneur charge for an apartment in order to maximize his revenue?

During a statistical survey, a public interest group obtains two estimates for the average monthly income of young adults aged 18 to 25. The first estimate is $$ 860$ and the second estimate is $$ 918$ . To refine its estimate, the group will take a weighted average of these two figures:

$I = 860 a + 918 (1 - a) where 0 \leq a \leq 1$

To get the best estimate, the group must choose $a$ to minimize the function

$V = 576 a^{2} + 5184 (1 - a)^{2}$

(The numbers that appear in this expression reflect the variance of the data, which measures how closely the data cluster around the mean, or average.) Find the value of $a$ that minimizes $V$ , and use this value to get a refined estimate for the average income.

$a equals 0.9$ ; $I equals $ 865.80$

The rate at which an antigen precipitates during an antigen-antibody reaction depends upon the amount of antigen present. For a fixed quantity of antibody, the time required for a particular antigen to precipitate is given in minutes by the function

$t = 2 w^{2} - 20 w + 54$

where $w$ is the quantity of antigen present, in grams. For what quantity of antigen will the reaction proceed most rapidly, and how long will the precipitation take?

For Problems 17–20, use transformations to graph the parabola. What is the vertex of each graph?

$y = (x - 3)^{2}$
$y = - (x - 3)^{2}$
$y = - (x - 3)^{2} + 4$

$y = (x + 1)^{2}$
$y = 2 (x + 1)^{2}$
$y = 2 (x + 1)^{2} - 4$

$y = (x + 4)^{2}$
$y = \frac{1}{2} (x + 4)^{2}$
$y = 3 + \frac{1}{2} (x + 4)^{2}$

$y = (x - 2)^{2}$
$y = - (x - 2)^{2}$
$y = - 3 - (x - 2)^{2}$

In Problems 21–24,

Find the vertex of the parabola.
Use transformations to sketch the graph.
Write the equation in standard form.

$y = 2 (x - 3)^{2} + 4$

$open parenthesis 3, 4 close parenthesis$
$y = 2 x^{2} - 12 x + 22$

$y = - 3 (x + 1)^{2} - 2$

$y = - \frac{1}{2} (x + 4)^{2} - 3$

$(- 4, - 3)$
$y = \frac{- 1}{2} x^{2} - 4 x - 11$

$y = 4 (x - 2)^{2} - 6$

For Problems 25–30,

Write each equation in the form $y = a (x - p)^{2} + q$ by completing the square.
Using horizontal and vertical translations, sketch the graph by hand.

$y = x^{2} - 4 x + 7$

$y = (x - 2)^{2} + 3$

$y = x^{2} - 2 x - 1$

$y = 3 x^{2} + 6 x - 2$

$y = 3 (x + 1)^{2} - 5$

$y = \frac{1}{2} x^{2} + 2 x + 5$

$y = - 2 x^{2} - 8 x + 3$

$y = - 2 (x + 2)^{2} + 11$

$y = - x^{2} + 5 x + 2$

A system of two quadratic equations may have no solution, one solution, or two solutions. Sketch a system illustrating each case. In your sketches, one of the parabolas should open up, and the other down.

No solutions:

One solution:

Two solutions:

two parabolas intersecting in two points

A system of two quadratic equations may have no solution, one solution, or two solutions. Sketch a system illustrating each case. In your sketches, both parabolas should open up.

For Problems 33–44, solve the system algebraically. Use your calculator to graph both equations and verify your solutions.

$\begin{aligned} y & = x^{2} - 4 x + 7 \\ y & = 11 - x \end{aligned}$

$open parenthesis minus 1, 12 close parenthesis, open parenthesis 4, 7 close parenthesis$

$\begin{aligned} y & = x^{2} + 6 x + 4 \\ y & = 3 x + 8 \end{aligned}$

$\begin{aligned} y & = - x^{2} - 2 x + 7 \\ y & = 2 x + 11 \end{aligned}$

$open parenthesis minus 2, 7 close parenthesis$

$\begin{aligned} y = x^{2} - 8 x + 17 \\ y + 4 x = 13 \end{aligned}$

$\begin{aligned} y = x^{2} + 8 x + 8 \\ 3 y + 2 x = - 36 \end{aligned}$

No solution

$\begin{aligned} y = - x^{2} + 4 x + 2 \\ 4 y - 3 x = 24 \end{aligned}$

$\begin{aligned} y & = x^{2} - 9 \\ y & = - 2 x^{2} + 9 x + 21 \end{aligned}$

$(- 2, - 5), (5, 16)$

$\begin{aligned} y & = 4 - x^{2} \\ y & = 3 x^{2} - 12 x - 12 \end{aligned}$

$\begin{aligned} y & = x^{2} - 0.5 x + 3.5 \\ y & = - x^{2} + 3.5 x + 1.5 \end{aligned}$

$open parenthesis 1, 4 close parenthesis$

$\begin{aligned} y & = x^{2} + 10 x + 22 \\ y & = - 0.5 x^{2} - 8 x - 32 \end{aligned}$

$\begin{aligned} y & = x^{2} - 4 x + 4 \\ y & = x^{2} - 8 x + 16 \end{aligned}$

$open parenthesis 3, 1 close parenthesis$

$\begin{aligned} y & = 0.5 x^{2} + 3 x + 5.5 \\ y & = 2 x^{2} + 12 x + 4 \end{aligned}$

Problems 45–48 deal with wildlife management and sustainable yield.

In Problem of Graphing Parabolas, you graphed the annual growth rate of a population of fish,

$R = f (x) = 0.4 x - 0.0001 x^{2}$

where $x$ is the current biomass of the population, in tons.

Suppose that fishermen harvest $300$ tons of fish each year. Sketch the graph of $H equals 300$ on the same axes with your graph of $y$ .
If the biomass is currently $2500$ tons and $300$ tons are harvested, will the population be larger or smaller next year? By how much? What if the biomass is currently $3500$ tons?
What sizes of biomass will remain stable from year to year if $300$ tons are harvested annually?
If the biomass ever falls below $1000$ tons, what will happen after several years of harvesting $300$ tons annually?

Larger, by $75$ tons. Smaller, by $125$ tons.
$1000$ tons and $3000$ tons
The fish population will decrease each year until it is completely depleted.

In Problem of Graphing Parabolas, you graphed the annual increase, $I$ , in the deer population in a national park,

$I = g (x) = 1.2 x - 0.0002 x^{2}$

where $x$ is the current population.

Suppose hunters are allowed to kill $1000$ deer per year. Sketch the graph of $H equals 1000$ on the same axes with a graph of $y$ .
What sizes of deer populations will remain stable from year to year if $1000$ deer are hunted annually?
Suppose $1600$ deer are killed annually. What sizes of deer populations will remain stable?
What is the largest annual harvest that still allows for a stable population? (This harvest is called the maximum sustainable yield.) What is the stable population?
What eventually happens if the population falls below the stable value but hunting continues at the maximum sustainable yield?

The annual increase, $N$ , in a bear population of size $x$ is given by

$N = F (x) = 0.0002 x (2000 - x)$

if the bears are not hunted. The number of bears killed each year by hunters is related to the bear population by the equation $K equals 0.2 x$ . (Notice that in this model, hunting is adjusted to the size of the bear population.)

Sketch the graphs of $N$ and $K$ on the same axes.
When the bear population is $1200$ , which is greater, $N$ or $K$ ? Will the population increase or decrease in the next year? By how many bears?
When the bear population is $900$ , will the population increase or decrease in the next year? By how many bears?
What sizes of bear population will remain stable after hunting?
What sizes of bear populations will increase despite hunting? What sizes of populations will decrease?
Toward what size will the population tend over time?
Suppose hunting limits are raised so that $K equals 0.3 x$ . Toward what size will the population tend over time?

$K > N$ . The population will decrease by $48$ bears.
The population will increase by $18$ bears.
$1000$
Populations between $0$ and $1000$ will increase; populations over $1000$ will decrease.
$1000$ (unless the population is $0$ )
$500$ (unless the population is $0$ )

The annual increase in the biomass of a whale population is given in tons by

$w = G (x) = 0.001 x (1000 - x)$

where $x$ is the current population, also in tons.

Sketch a graph of $w$ for $0 less than or equal to x less than or equal to 1100$ . What size biomass remains stable?
Each year hunters are allowed to harvest a biomass given by $H equals 0.6 x$ . Sketch $H$ on the same graph with $w$ . What is the stable biomass with hunting?
What sizes of populations will increase despite hunting? What sizes will decrease?
What size will the population approach over time? What biomass are hunters allowed to harvest for that size population?
Find a value of $k$ so that the graph of $H = k x$ will pass through the vertex of $w = 0.001 x (1000 - x)$ .
For the value of $k$ found in part (e), what size will the population approach over time? What biomass are hunters allowed to harvest for that size population?
Explain why the whaling industry should prefer hunting quotas of $k x$ rather than $0.6 x$ for a long-term strategy, even though $0.6 x greater than k x$ for any positive value of $x$ .

For Problems 49–52,

Find the break-even points by solving a system of equations.
Graph the equations for Revenue and Cost in the same window and verify your solutions on the graph.
Use the fact that
$Profit = Revenue - Cost$
to find the value of $x$ for which profit is maximum.

Writewell, Inc. makes fountain pens. It costs Writewell

$C = 0.0075 x^{2} + x + 2100$

dollars to manufacture $x$ pens, and the company receives $R equals 13 x$ dollars in revenue from the sale of the pens.

$open parenthesis 200, 2600 close parenthesis, open parenthesis 1400, 18, 200 close parenthesis$
$x equals 800$

It costs The Sweetshop

$C = 0.01 x^{2} + 1836$

dollars to produce $x$ pounds of chocolate creams. The company brings in $R equals 12 x$ dollars revenue from the sale of the chocolates.

It costs an appliance manufacturer

$C = 1.625 x^{2} + 33, 150$

dollars to produce $x$ front-loading washing machines, which will then bring in revenues of $R equals 650 x$ dollars.

$open parenthesis 60, 39, 000 close parenthesis, open parenthesis 340, 221, 000 close parenthesis$
$x equals 200$

A company can produce $x$ lawn mowers for a cost of

$C = 0.125 x^{2} + 100, 000$

dollars. The sale of the lawn mowers will generate $R equals 300 x$ dollars in revenue.

Problems 53 and 54 prove that the vertical line $x = \frac{- b}{2 a}$ is the axis of symmetry of the graph of $f (x) = a x^{2} + b x + c$ . A graph is symmetric about the line $x = h$ if the point $(h + d, v)$ lies on the graph whenever the point $(h - d, v)$ lies on the graph.

Sketch a parabola $f (x) = a (x - h)^{2} + k$ and the line $x = h$ . We will show that the parabola is symmetric about the line $x = h$ .
Label a point on the parabola with $x$ -coordinate $x = h + d$ , where $d greater than 0$ . What is the $y$ -coordinate of that point?
Label the point on the parabola with $x$ -coordinate $x = h - d$ . What is the $y$ -coordinate of that point?
Explain why your answers to parts (b) and (c) prove that the line $x = h$ is the axis of symmetry for the graph of $f (x) = a (x - h)^{2} + k$ .

See graph and (c)
$a d^{2} + k$
The two points on the parabola that are the same horizontal distance from the line $x = h$ the axis of symmetry have the same $y$ -coordinate, so they are symmetric about that line.

To find the axis of symmetry for the graph of $g (x) = a x^{2} + b x + c$ , we will use the results of Problem 51 and the technique of completing the square.

Write the equation $y = a x^{2} + b x + c$ in vertex form by completing the square. (Follow the steps in Example.)
Your answer to part (a) has the form $y = a (x - h)^{2} + k$ . What is your value of $h$ ? What is your value of $k$ ?
What is the axis of symmetry for the parabola $g (x) = a x^{2} + b x + c$ ?

Modeling, Functions, and Graphs by Katherine Yoshiwara (yoshiwarabooks.org), GNU Free Documentation License 1.2 or later. Adapted for the XYZ HTML edition with the authors' permission (recorded 2026-07-04). License: GFDL-1.2-or-later.

Number of price reductions	Price of ticket	Number of tickets sold	Total revenue
$0$	$5.00$	$100$	$500$
$1$	$4.75$	$110$	$522.50$
$2$
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$12$

No. of price increases	Price of tape	No. of tapes sold	Total revenue
$0$	$6$	$96$	$576$
$1$	$6.50$	$92$	$598$
$2$
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$4$
$5$
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$12$
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$24$