11.2 Parabolas

Graph Vertical Parabolas

The next conic section we will look at is a parabola. We define a parabola as all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.

This figure shows a double cone. The bottom nappe is intersected by a plane in such a way that the intersection forms a parabola.

Previously, we learned to graph vertical parabolas from the general form or the standard form using properties. Those methods will also work here. We will summarize the properties here.

Vertical Parabolas
	General form $y = a x^{2} + b x + c$	Standard form $y = a {(x - h)}^{2} + k$
Orientation	$a greater than 0$ up; $a less than 0$ down	$a greater than 0$ up; $a less than 0$ down
Axis of symmetry	$x = - \frac{b}{2 a}$	$x = h$
Vertex	Substitute $x = - \frac{b}{2 a}$ and solve for y.	$(h, k)$
y-intercept	Let $x equals 0$	Let $x equals 0$
x-intercepts	Let $y equals 0$	Let $y equals 0$

The graphs show what the parabolas look like when they open up or down. Their position in relation to the x- or y-axis is merely an example.

This figure shows two parabolas with axis x equals h and vertex h, k. The one on the left opens up and A is greater than 0. The one on the right opens down. Here A is less than 0.

To graph a parabola from these forms, we used the following steps.

The next example reviews the method of graphing a parabola from the general form of its equation.

Example 1

Graph $y = − x^{2} + 6 x - 8$ by using properties.


Since a is $−1,$ the parabola opens downward.

To find the axis of symmetry, find $x = - \frac{b}{2 a} .$


	The axis of symmetry is $x equals 3.$

The vertex is on the line $x equals 3.$
Let $x equals 3.$


	The vertex is $open parenthesis 3, 1 close parenthesis.$

The y-intercept occurs when $x equals 0.$
Substitute $x equals 0.$
Simplify.
	The y-intercept is $open parenthesis 0, −8 close parenthesis.$
The point $open parenthesis 0, −8 close parenthesis$ is three units to the left of the line of symmetry. The point three units to the right of the line of symmetry is $open parenthesis 6, −8 close parenthesis.$	Point symmetric to the y-intercept is $open parenthesis 6, −8 close parenthesis.$

The x-intercept occurs when $y equals 0.$
Let $y equals 0.$
Factor the GCF.
Factor the trinomial.
Solve for x.
	The x-intercepts are $open parenthesis 4, 0 close parenthesis, open parenthesis 2, 0 close parenthesis.$
Graph the parabola.

The next example reviews the method of graphing a parabola from the standard form of its equation, $y = a {(x - h)}^{2} + k .$

Example 2

Write $y = 3 x^{2} - 6 x + 5$ in standard form and then use properties of standard form to graph the equation.

Rewrite the function in $y = a {(x - h)}^{2} + k$ form by completing the square.	$y = 3 x^{2} - 6 x + 5$
	$y = 3 (x^{2} - 2 x) + 5$
	$y = 3 (x^{2} - 2 x + 1) + 5 - 3$
	$y = 3 {(x - 1)}^{2} + 2$
Identify the constants a, h, k.	$a equals 3$ , $h equals 1$ , $k equals 2$
Since $a equals 3,$ the parabola opens upward.

The axis of symmetry is $x = h .$	The axis of symmetry is $x equals 1.$
The vertex is $(h, k) .$	The vertex is $open parenthesis 1, 2 close parenthesis.$
Find the y-intercept by substituting $x equals 0.$	$y = 3 {(x - 1)}^{2} + 2$ $y = 3 \cdot 0^{2} - 6 \cdot 0 + 5$
	$y equals 5$
	y-intercept $open parenthesis 0, 5 close parenthesis$
Find the point symmetric to $open parenthesis 0, 5 close parenthesis$ across the axis of symmetry.	$open parenthesis 2, 5 close parenthesis$
Find the x-intercepts.	$\begin{matrix} y & = & 3 {(x - 1)}^{2} + 2 \\ 0 & = & 3 {(x - 1)}^{2} + 2 \\ - 2 & = & 3 {(x - 1)}^{2} \\ - \frac{2}{3} & = & {(x - 1)}^{2} \\ \pm \sqrt{- \frac{2}{3}} & = & x - 1 \end{matrix}$
	The square root of a negative number tells us the solutions are complex numbers. So there are no x-intercepts.
Graph the parabola.

Graph Horizontal Parabolas

Our work so far has only dealt with parabolas that open up or down. We are now going to look at horizontal parabolas. These parabolas open either to the left or to the right. If we interchange the x and y in our previous equations for parabolas, we get the equations for the parabolas that open to the left or to the right.

**Table 11.1**
Horizontal Parabolas
	General form $x = a y^{2} + b y + c$	Standard form $x = a {(y - k)}^{2} + h$
Orientation	$a greater than 0$ right; $a less than 0$ left	$a greater than 0$ right; $a less than 0$ left
Axis of symmetry	$y = - \frac{b}{2 a}$	$y = k$
Vertex	Substitute $y = - \frac{b}{2 a}$ and solve for x.	$(h, k)$
y-intercepts	Let $x equals 0$	Let $x equals 0$
x-intercept	Let $y equals 0$	Let $y equals 0$

The graphs show what the parabolas look like when they to the left or to the right. Their position in relation to the x- or y-axis is merely an example.

This figure shows two parabolas with axis of symmetry y equals k,) and vertex (h, k. The one on the left is labeled a greater than 0 and opens to the right. The other parabola opens to the left.

Looking at these parabolas, do their graphs represent a function? Since both graphs would fail the vertical line test, they do not represent a function.

To graph a parabola that opens to the left or to the right is basically the same as what we did for parabolas that open up or down, with the reversal of the x and y variables.

Example 3

Graph $x equals 2 y squared$ by using properties.


Since $a equals 2,$ the parabola opens to the right.

To find the axis of symmetry, find $y = - \frac{b}{2 a} .$


	The axis of symmetry is $y equals 0.$
The vertex is on the line $y equals 0.$
Let $y equals 0.$

	The vertex is $open parenthesis 0, 0 close parenthesis.$

Since the vertex is $open parenthesis 0, 0 close parenthesis,$ both the x- and y-intercepts are the point $open parenthesis 0, 0 close parenthesis.$ To graph the parabola we need more points. In this case it is easiest to choose values of y.

We also plot the points symmetric to

open parenthesis 2, 1 close parenthesis

and

open parenthesis 8, 2 close parenthesis

across the y-axis, the points

open parenthesis 2, −1 close parenthesis,

open parenthesis 8, −2 close parenthesis.

Graph the parabola.

In the next example, the vertex is not the origin.

Example 4

Graph $x = − y^{2} + 2 y + 8$ by using properties.


Since $a equals −1,$ the parabola opens to the left.

To find the axis of symmetry, find $y = - \frac{b}{2 a} .$


	The axis of symmetry is $y equals 1.$
The vertex is on the line $y equals 1.$
Let $y equals 1.$

	The vertex is $open parenthesis 9, 1 close parenthesis.$
The x-intercept occurs when $y equals 0.$


	The x-intercept is $open parenthesis 8, 0 close parenthesis.$
The point $open parenthesis 8, 0 close parenthesis$ is one unit below the line of symmetry. The symmetric point one unit above the line of symmetry is $open parenthesis 8, 2 close parenthesis$	Symmetric point is $open parenthesis 8, 2 close parenthesis.$
The y-intercept occurs when $x equals 0.$
Substitute $x equals 0.$
Solve.


	The y-intercepts are $open parenthesis 0, 4 close parenthesis$ and $open parenthesis 0, −2 close parenthesis.$
Connect the points to graph the parabola.

In Table 11.1, we see the relationship between the equation in standard form and the properties of the parabola. The How To box lists the steps for graphing a parabola in the standard form $x = a {(y - k)}^{2} + h .$ We will use this procedure in the next example.

Example 5

Graph $x = 2 {(y - 2)}^{2} + 1$ using properties.


Identify the constants a, h, k.	$a equals 2,$ $h equals 1,$ $k equals 2$
Since $a equals 2,$ the parabola opens to the right.

The axis of symmetry is $y = k .$	The axis of symmetry is $y equals 2.$
The vertex is $(h, k) .$	The vertex is $open parenthesis 1, 2 close parenthesis.$
Find the x-intercept by substituting $y equals 0.$	$\begin{matrix} x & = & 2 {(y - 2)}^{2} + 1 \\ x & = & 2 {(0 - 2)}^{2} + 1 \\ x & = & 9 \end{matrix}$
	The x-intercept is $open parenthesis 9, 0 close parenthesis.$
Find the point symmetric to $open parenthesis 9, 0 close parenthesis$ across the axis of symmetry.	$open parenthesis 9, 4 close parenthesis$
Find the y-intercepts. Let $x equals 0.$	$\begin{matrix} x & = & 2 {(y - 2)}^{2} + 1 \\ 0 & = & 2 {(y - 2)}^{2} + 1 \\ −1 & = & 2 {(y - 2)}^{2} \end{matrix}$
	A square cannot be negative, so there is no real solution. So there are no y-intercepts.
Graph the parabola.

In the next example, we notice the a is negative and so the parabola opens to the left.

Example 6

Graph $x = −4 {(y + 1)}^{2} + 4$ using properties.


Identify the constants a, h, k.	$a equals −4,$ $h equals 4,$ $k equals −1$
Since $a equals −4,$ the parabola opens to the left.

The axis of symmetry is $y = k .$	The axis of symmetry is $y equals −1.$
The vertex is $(h, k) .$	The vertex is $open parenthesis 4, −1 close parenthesis.$
Find the x-intercept by substituting $y equals 0.$	$\begin{matrix} x & = & −4 {(y + 1)}^{2} + 4 \\ x & = & −4 {(0 + 1)}^{2} + 4 \\ x & = & 0 \end{matrix}$
	The x-intercept is $open parenthesis 0, 0 close parenthesis.$
Find the point symmetric to $open parenthesis 0, 0 close parenthesis$ across the axis of symmetry.	$open parenthesis 0, −2 close parenthesis$
Find the y-intercepts.	$x = −4 {(y + 1)}^{2} + 4$
Let $x equals 0.$	$\begin{matrix} 0 & = & −4 {(y + 1)}^{2} + 4 \\ −4 & = & −4 {(y + 1)}^{2} \\ 1 & = & {(y + 1)}^{2} \\ y + 1 & = & \pm 1 \end{matrix}$
	$y = −1 + 1 y = −1 - 1$
	$y = 0 y = −2$
	The y-intercepts are $open parenthesis 0, 0 close parenthesis$ and $open parenthesis 0, −2 close parenthesis.$
Graph the parabola.

The next example requires that we first put the equation in standard form and then use the properties.

Example 7

Write $x = 2 y^{2} + 12 y + 17$ in standard form and then use the properties of the standard form to graph the equation.


Rewrite the function in $x = a {(y - k)}^{2} + h$ form by completing the square.



Identify the constants a, h, k.	$a = 2, h = −1, k = −3$
Since $a equals 2,$ the parabola opens to the right.

The axis of symmetry is $y = k .$	The axis of symmetry is $y equals −3.$
The vertex is $(h, k) .$	The vertex is $open parenthesis −1, −3 close parenthesis.$
Find the x-intercept by substituting $y equals 0.$	$\begin{matrix} x & = & 2 {(y + 3)}^{2} - 1 \\ x & = & 2 {(0 + 3)}^{2} - 1 \\ x & = & 17 \end{matrix}$
	The x-intercept is $open parenthesis 17, 0 close parenthesis.$
Find the point symmetric to $open parenthesis 17, 0 close parenthesis$ across the axis of symmetry.	$open parenthesis 17, −6 close parenthesis$
Find the y-intercepts. Let $x equals 0.$	$\begin{matrix} x & = & 2 {(y + 3)}^{2} - 1 \\ 0 & = & 2 {(y + 3)}^{2} - 1 \\ 1 & = & 2 {(y + 3)}^{2} \\ \frac{1}{2} & = & {(y + 3)}^{2} \\ y + 3 & = & \pm \sqrt{\frac{1}{2}} \\ y & = & −3 \pm \frac{\sqrt{2}}{2} \end{matrix}$
	$y = −3 + \frac{\sqrt{2}}{2} y = −3 - \frac{\sqrt{2}}{2}$
	$y \approx - 2.3 y \approx - 3.7$
	The y-intercepts are $(0, −3 + \frac{\sqrt{2}}{2}), (0, −3 - \frac{\sqrt{2}}{2}) .$
Graph the parabola.

Solve Applications with Parabolas

Many architectural designs incorporate parabolas. It is not uncommon for bridges to be constructed using parabolas as we will see in the next example.

Example 8

Find the equation of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form.

We will first set up a coordinate system and draw the parabola. The graph will give us the information we need to write the equation of the graph in the standard form $y = a {(x - h)}^{2} + k .$

Let the lower left side of the bridge be the origin of the coordinate grid at the point $open parenthesis 0, 0 close parenthesis.$ Since the base is 20 feet wide the point $open parenthesis 20, 0 close parenthesis$ represents the lower right side. The bridge is 10 feet high at the highest point. The highest point is the vertex of the parabola so the y-coordinate of the vertex will be 10. Since the bridge is symmetric, the vertex must fall halfway between the left most point, $open parenthesis 0, 0 close parenthesis,$ and the rightmost point $open parenthesis 20, 0 close parenthesis.$ From this we know that the x-coordinate of the vertex will also be 10.
Identify the vertex, $(h, k) .$	$open parenthesis h, k close parenthesis equals open parenthesis 10, 10 close parenthesis$
	$h = 10, k = 10$
Substitute the values into the standard form. The value of a is still unknown. To find the value of a use one of the other points on the parabola.	$\begin{matrix} y & = & a {(x - h)}^{2} + k \\ y & = & a {(x - 10)}^{2} + 10 \\ (x, y) & = & (0, 0) \end{matrix}$
Substitute the values of the other point into the equation.	$\begin{matrix} y & = & a {(x - 10)}^{2} + 10 \\ 0 & = & a {(0 - 10)}^{2} + 10 \end{matrix}$
Solve for a.	$\begin{matrix} 0 & = & a {(0 - 10)}^{2} + 10 \\ −10 & = & a {(−10)}^{2} \\ −10 & = & 100 a \\ \frac{−10}{100} & = & a \\ a & = & - \frac{1}{10} \end{matrix}$
	$y = a {(x - 10)}^{2} + 10$
Substitute the value for a into the equation.	$y = - \frac{1}{10} {(x - 10)}^{2} + 10$

Key Concepts

Parabola: A parabola is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.

Vertical Parabolas
	General form $y = a x^{2} + b x + c$	Standard form $y = a {(x - h)}^{2} + k$
Orientation	$a greater than 0$ up; $a less than 0$ down	$a greater than 0$ up; $a less than 0$ down
Axis of symmetry	$x = - \frac{b}{2 a}$	$x = h$
Vertex	Substitute $x = - \frac{b}{2 a}$ and solve for y.	$(h, k)$
y- intercept	Let $x equals 0$	Let $x equals 0$
x-intercepts	Let $y equals 0$	Let $y equals 0$

How to graph vertical parabolas $(y = a x^{2} + b x + c$ or $f (x) = a {(x - h)}^{2} + k)$ using properties.

Determine whether the parabola opens upward or downward.
Find the axis of symmetry.
Find the vertex.
Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry.
Find the x-intercepts.
Graph the parabola.

Horizontal Parabolas
	General form $x = a y^{2} + b y + c$	Standard form $x = a {(y - k)}^{2} + h$
Orientation	$a greater than 0$ right; $a less than 0$ left	$a greater than 0$ right; $a less than 0$ left
Axis of symmetry	$y = - \frac{b}{2 a}$	$y = k$
Vertex	Substitute $y = - \frac{b}{2 a}$ and solve for x.	$(h, k)$
y-intercepts	Let $x equals 0$	Let $x equals 0$
x-intercept	Let $y equals 0$	Let $y equals 0$

How to graph horizontal parabolas $(x = a y^{2} + b y + c$ or $x = a {(y - k)}^{2} + h)$ using properties.
1. Determine whether the parabola opens to the left or to the right.
2. Find the axis of symmetry.
3. Find the vertex.
4. Find the x-intercept. Find the point symmetric to the x-intercept across the axis of symmetry.
5. Find the y-intercepts.
6. Graph the parabola.

Practice Makes Perfect

Graph Vertical Parabolas

In the following exercises, graph each equation by using properties.

$y = − x^{2} + 4 x - 3$

This graph shows a parabola opening downward with vertex (2, 1) and x intercepts (1, 0) and (3, 0).

$y = − x^{2} + 8 x - 15$

$y = 6 x^{2} + 2 x - 1$

This graph shows a parabola opening upward. The vertex is (negative 0.167, negative 1.167), the x intercepts are (negative 0.608) and (negative 0.274, 0), and the y-intercept is (0, negative 1).

$y = 8 x^{2} - 10 x + 3$

In the following exercises, ⓐ write the equation in standard form and ⓑ use properties of the standard form to graph the equation.

$y = − x^{2} + 2 x - 4$

ⓐ $y = − {(x - 1)}^{2} - 3$
ⓑ

This graph shows a parabola opening downward with vertex (1, negative 3) and y intercept (0, 4).

$y = 2 x^{2} + 4 x + 6$

$y = −2 x^{2} - 4 x - 5$

ⓐ $y = −2 {(x + 1)}^{2} - 3$
ⓑ

This graph shows a parabola opening downward with vertex (negative 1, negative 3) and x intercepts (negative 5, 0).

$y = 3 x^{2} - 12 x + 7$

Graph Horizontal Parabolas

In the following exercises, graph each equation by using properties.

$x equals −2 y squared$

This graph shows a parabola opening to the left with vertex (0, 0). Two points on it are (negative 2, 1) and (negative 2, negative 1).

$x equals 3 y squared$

$x equals 4 y squared$

This graph shows a parabola opening to the right with vertex (0, 0). Two points on it are (4, 1) and (4, negative 1).

$x equals −4 y squared$

$x = − y^{2} - 2 y + 3$

This graph shows a parabola opening to the left with vertex (4, negative 1) and y intercepts (0, 1) and (0, negative 3).

$x = − y^{2} - 4 y + 5$

$x = y^{2} + 6 y + 8$

This graph shows a parabola opening to the right with vertex (negative 1, negative 3) and y intercepts (0, negative 2) and (0, negative 4).

$x = y^{2} - 4 y - 12$

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

This graph shows a parabola opening to the right with vertex (3, 2) and x intercept (7, 0).

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

$x = − {(y - 1)}^{2} + 2$

This graph shows a parabola opening to the left with vertex (2, 1) and x intercept (1, 0).

$x = − {(y - 4)}^{2} + 3$

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

This graph shows a parabola opening to the right with vertex (1, negative 2) and x intercept (5, 0).

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

$x = − {(y + 3)}^{2} + 2$

This graph shows a parabola opening to the left with vertex (2, negative 3). Two points on it are (negative 2, negative 1) and (negative 2, 5).

$x = − {(y + 4)}^{2} + 3$

$x = −3 {(y - 2)}^{2} + 3$

This graph shows a parabola opening to the left with vertex (3, 2) and y intercepts (0, 1) and (0, 3).

$x = −2 {(y - 1)}^{2} + 2$

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

This graph shows a parabola opening to the right with vertex (negative 4, negative 1) and y intercepts (0, 0) and (0, negative 2).

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

In the following exercises, ⓐ write the equation in standard form and ⓑ use properties of the standard form to graph the equation.

$x = y^{2} + 4 y - 5$

ⓐ $x = {(y + 2)}^{2} - 9$
ⓑ

This graph shows a parabola opening to the right with vertex (negative 9, negative 2) and y intercepts (0, 1) and (0, negative 5).

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

$x = −2 y^{2} - 12 y - 16$

ⓐ $x = −2 {(y + 3)}^{2} + 2$
ⓑ

This graph shows a parabola opening to the left with vertex (2, negative 3) and y intercepts (0, negative 2) and (0, negative 4).

$x = −3 y^{2} - 6 y - 5$

Mixed Practice

In the following exercises, match each graph to one of the following equations: ⓐ x² + y² = 64 ⓑ x² + y² = 49
ⓒ (x + 5)² + (y + 2)² = 4 ⓓ (x − 2)² + (y − 3)² = 9 ⓔ y = −x² + 8x − 15 ⓕ y = 6x² + 2x − 1

This graph shows circle with center (0, 0) and radius 8 units.

ⓐ

This graph shows a parabola opening upwards. Its vertex has an x value of slightly less than 0 and a y value of slightly less than negative 1. A point on it is close to (negative 1, 3).

This graph shows circle with center (0, 0) and radius 7 units.

ⓑ

This graph shows a parabola opening downwards with vertex (4, 1) and x intercepts (3, 0) and (5, 0).

This graph shows circle with center (2, 3) and radius 3 units.

ⓓ

This graph shows circle with center (negative 5, negative 2) and radius 2 units.

Solve Applications with Parabolas

Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form.

$y = - \frac{1}{15} {(x - 15)}^{2} + 15$

Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form.

This figure shows a parabolic arch formed in the foundation of a bridge. It is 50 feet high and 100 feet wide at the base.

Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form.

This figure shows a parabolic arch formed in the foundation of a bridge. It is 90 feet high and 60 feet wide at the base.

$y = - \frac{1}{10} {(x - 30)}^{2} + 90$

Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form.

This figure shows a parabolic arch formed in the foundation of a bridge. It is 45 feet high and 30 feet wide at the base.

Writing Exercises

In your own words, define a parabola.

Answers will vary.

Is the parabola $y = x^{2}$ a function? Is the parabola $x = y^{2}$ a function? Explain why or why not.

Write the equation of a parabola that opens up or down in standard form and the equation of a parabola that opens left or right in standard form. Provide a sketch of the parabola for each one, label the vertex and axis of symmetry.

Answers will vary.

Explain in your own words, how you can tell from its equation whether a parabola opens up, down, left or right.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has four columns, 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don’t get it. The first column has the following statements: graph vertical parabolas, graph horizontal parabolas, solve applications with parabolas. The remaining columns are blank.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?