Conics

# Parabolas

### Learning Objectives

By the end of this section, you will be able to:

• Graph vertical parabolas
• Graph horizontal parabolas
• Solve applications with parabolas

Before you get started, take this readiness quiz.

1. Graph:
If you missed this problem, review (Figure).
2. Solve by completing the square:
If you missed this problem, review (Figure).
3. Write in standard form:
If you missed this problem, review (Figure).

### 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.

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.

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
Standard form
Orientation up; down up; down
Axis of symmetry
Vertex Substitute and
solve for y.
y-intercept Let Let
x-intercepts Let Let

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.

Graph vertical parabolas using properties.
1. Determine whether the parabola opens upward or downward.
2. Find the axis of symmetry.
3. Find the vertex.
4. Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry.
5. Find the x-intercepts.
6. Graph the parabola.

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

Graph by using properties.

 Since a is the parabola opens downward. To find the axis of symmetry, find The axis of symmetry is The vertex is on the line Let The vertex is The y-intercept occurs when Substitute Simplify. The y-intercept is The point is three units to the left of theline of symmetry. The point three units to theright of the line of symmetry is Point symmetric to the y-intercept is The x-intercept occurs when Let Factor the GCF. Factor the trinomial. Solve for x. The x-intercepts are Graph the parabola.

Graph by using properties.

Graph by using properties.

The next example reviews the method of graphing a parabola from the standard form of its equation,

Write in standard form and then use properties of standard form to graph the equation.

 Rewrite the function in formby completing the square. Identify the constants a, h, k. , , Since the parabola opens upward. The axis of symmetry is The axis of symmetry is The vertex is The vertex is Find the y-intercept by substituting y-intercept Find the point symmetric to across the axis of symmetry. Find the x-intercepts. The square root of a negative numbertells us the solutions are complexnumbers. So there are no x-intercepts. Graph the parabola.

Write in standard form and use properties of standard form to graph the equation.

Write in standard form and use properties of standard form to graph the equation.

### 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.

Horizontal Parabolas
General form
Standard form
Orientation right; left right; left
Axis of symmetry
Vertex Substitute and
solve for x.
y-intercepts Let Let
x-intercept Let Let

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.

Graph horizontal parabolas 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.

Graph by using properties.

 Since the parabola opens to the right. To find the axis of symmetry, find The axis of symmetry is The vertex is on the line Let The vertex is

Since the vertex is both the x– and y-intercepts are the point 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 and across the y-axis, the points

Graph the parabola.

Graph by using properties.

Graph by using properties.

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

Graph by using properties.

 Since the parabola opens to the left. To find the axis of symmetry, find The axis of symmetry is The vertex is on the line Let The vertex is The x-intercept occurs when The x-intercept is The point is one unit below the line ofsymmetry. The symmetric point one unitabove the line of symmetry is Symmetric point is The y-intercept occurs when Substitute Solve. The y-intercepts are and Connect the points to graph the parabola.

Graph by using properties.

Graph by using properties.

In (Figure), 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 We will use this procedure in the next example.

Graph using properties.

 Identify the constants a, h, k. Since the parabola opens to the right. The axis of symmetry is The axis of symmetry is The vertex is The vertex is Find the x-intercept by substituting The x-intercept is Find the point symmetric to across theaxis of symmetry. Find the y-intercepts. Let A square cannot be negative, so there is no realsolution. So there are no y-intercepts. Graph the parabola.

Graph using properties.

Graph using properties.

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

Graph using properties.

 Identify the constants a, h, k. Since the parabola opens to the left. The axis of symmetry is The axis of symmetry is The vertex is The vertex is Find the x-intercept by substituting The x-intercept is Find the point symmetric to across theaxis of symmetry. Find the y-intercepts. Let The y-intercepts are and Graph the parabola.

Graph using properties.

Graph using properties.

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

Write in standard form and then use the properties of the standard form to graph the equation.

 Rewrite the function in form by completingthe square. Identify the constants a, h, k. Since the parabola opens tothe right. The axis of symmetry is The axis of symmetry is The vertex is The vertex is Find the x-intercept by substituting The x-intercept is Find the point symmetric to across the axis of symmetry. Find the y-intercepts.Let The y-intercepts are Graph the parabola.

Write in standard form and use properties of the standard form to graph the equation.

Write in standard form and use properties of the standard form to graph the equation.

### 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.

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

 Let the lower left side of the bridge be theorigin of the coordinate grid at the point Since the base is 20 feet wide the point represents the lower right side.The bridge is 10 feet high at the highestpoint. The highest point is the vertex ofthe parabola so the y-coordinate of thevertex will be 10.Since the bridge is symmetric, the vertexmust fall halfway between the left mostpoint, and the rightmost point From this we know that thex-coordinate of the vertex will also be 10. Identify the vertex, Substitute the values into the standard form.The value of a is still unknown. To findthe value of a use one of the other pointson the parabola. Substitute the values of the other pointinto the equation. Solve for a. Substitute the value for a into theequation.

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

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

Access these online resources for additional instructions and practice with quadratic functions and parabolas.

### 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
Standard form
Orientation up; down up; down
Axis of symmetry
Vertex Substitute and
solve for y.
y– intercept Let Let
x-intercepts Let Let

• How to graph vertical parabolas or using properties.
1. Determine whether the parabola opens upward or downward.
2. Find the axis of symmetry.
3. Find the vertex.
4. Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry.
5. Find the x-intercepts.
6. Graph the parabola.

Horizontal Parabolas
General form
Standard form
Orientation right; left right; left
Axis of symmetry
Vertex Substitute and
solve for x.
y-intercepts Let Let
x-intercept Let Let

• How to graph horizontal parabolas or 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.

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: 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

Solve Applications with Parabolas

Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Use the lower left side of the bridge as the origin (0, 0).

Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Use the lower left side of the bridge as the origin (0, 0).

Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Use the lower left side of the bridge as the origin (0, 0).

Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Use the lower left side of the bridge as the origin (0, 0).

#### Writing Exercises

In your own words, define a parabola.

Is the parabola a function? Is the parabola 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.

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.

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

### Glossary

parabola
A parabola is all points in a plane that are the same distance from a fixed point and a fixed line.