11.2 Parabolas

Learning objectives

Learning objectives

• Graph vertical parabolas

• Graph horizontal parabolas

• Solve applications with 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.

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.

General form$y=a{x}^{2}+bx+c$ · Standard form$y=a{(x-h)}^{2}+k$ Orientation · $a>0$ up; $a<0$ down · $a>0$ up; $a<0$ down Axis of symmetry · $x=-\frac{b}{2a}$ · $x=h$ Vertex · Substitute $x=-\frac{b}{2a}$ andsolve for y. · $(h,k)$ y-intercept · Let $x=0$ · Let $x=0$ x-intercepts · Let $y=0$ · Let $y=0$

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.

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 11.12

Solution

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

Example 11.13

Solution

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.

General form$x=a{y}^{2}+by+c$ · Standard form$x=a{(y-k)}^{2}+h$ Orientation · $a>0$ right; $a<0$ left · $a>0$ right; $a<0$ left Axis of symmetry · $y=-\frac{b}{2a}$ · $y=k$ Vertex · Substitute $y=-\frac{b}{2a}$ andsolve for x. · $(h,k)$ y-intercepts · Let $x=0$ · Let $x=0$ x-intercept · Let $y=0$ · Let $y=0$

Table

11.1

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.

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 11.14

Solution

Since the vertex is $(0,0),$ both the x- and y-intercepts are the point $(0,0).$ 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 $(2,1)$ and $(8,2)$ across the y-axis, the points $(2,-1),$$(8,-2).$

Graph the parabola.

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

Example 11.15

Solution

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 11.16

Solution

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

Example 11.17

Solution

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

Example 11.18

Solution

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 11.19

Solution

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$y=a{(x-h)}^{2}+k.$

Section 11.2 Exercises

Section 11.2 Exercises

Graph Vertical Parabolas

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

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

Graph Horizontal Parabolas

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

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

Mixed Practice

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

Solve Applications with Parabolas

Writing Exercises

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

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

1. Graph $y=-{x}^{2}+5x-6$ by using properties.

2. Graph $y=-{x}^{2}+8x-12$ by using properties.

3. ⓐ Write $y=2{x}^{2}+4x+5$ in standard form and ⓑ use properties of standard form to graph the equation.

1. ⓐ Write $y=-2{x}^{2}+8x-7$ in standard form and ⓑ use properties of standard form to graph the equation.

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

3. Graph $x=-{y}^{2}$ by using properties.

1. Graph $x=-{y}^{2}-4y+12$ by using properties.

2. Graph $x=-{y}^{2}+2y-3$ by using properties.

3. Graph $x=3{(y-1)}^{2}+2$ using properties.

1. Graph $x=2{(y-3)}^{2}+2$ using properties.

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

3. Graph $x=-2{(y+3)}^{2}+2$ using properties.

1. ⓐ Write $x=3{y}^{2}+6y+7$ in standard form and ⓑ use properties of the standard form to graph the equation.

2. ⓐ Write $x=-4{y}^{2}-16y-12$ in standard form and ⓑ use properties of the standard form to graph the equation.

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

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

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

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

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

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

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

1. $x=4{y}^{2}$

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

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

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

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

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

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

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

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

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

2. $x=-2{y}^{2}-12y-16$

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

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

2. In your own words, define a parabola.

3. 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.