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Intermediate Algebra

11.2 Parabolas

Intermediate Algebra11.2 Parabolas
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  15. Index

Learning Objectives

By the end of this section, you will be able to:
  • Graph vertical parabolas
  • Graph horizontal parabolas
  • Solve applications with parabolas
Be Prepared 11.2

Before you get started, take this readiness quiz.

  1. Graph: y=−3x2+12x12.y=−3x2+12x12.
    If you missed this problem, review Example 9.47.
  2. Solve by completing the square: x26x+6=0.x26x+6=0.
    If you missed this problem, review Example 9.12.
  3. Write in standard form: y=3x26x+5.y=3x26x+5.
    If you missed this problem, review Example 9.59.

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.

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.

This figure shows a parabola opening upwards. Below the parabola is a horizontal line labeled directrix. A vertical dashed line through the center of the parabola is labeled axis of symmetry. The point where the axis intersects the parabola is labeled vertex. A point on the axis, within the parabola is labeled focus. A line perpendicular to the directrix connects the directrix to a point on the parabola and another line connects this point to the focus. Both these lines are of the same length.

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=ax2+bx+cy=ax2+bx+c
Standard form
y=a(xh)2+ky=a(xh)2+k
Orientation a>0a>0 up; a<0a<0 down a>0a>0 up; a<0a<0 down
Axis of symmetry x=b2ax=b2a x=hx=h
Vertex Substitute x=b2ax=b2a and
solve for y.
(h,k)(h,k)
y-intercept Let x=0x=0 Let x=0x=0
x-intercepts Let y=0y=0 Let y=0y=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.

How To

Graph vertical parabolas (y=ax2+bx+corf(x)=a(xh)2+k)(y=ax2+bx+corf(x)=a(xh)2+k) using properties.

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

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

Example 11.12

Graph y=x2+6x8y=x2+6x8 by using properties.

Try It 11.23

Graph y=x2+5x6y=x2+5x6 by using properties.

Try It 11.24

Graph y=x2+8x12y=x2+8x12 by using properties.

The next example reviews the method of graphing a parabola from the standard form of its equation, y=a(xh)2+k.y=a(xh)2+k.

Example 11.13

Writey=3x26x+5y=3x26x+5 in standard form and then use properties of standard form to graph the equation.

Try It 11.25

Write y=2x2+4x+5y=2x2+4x+5 in standard form and use properties of standard form to graph the equation.

Try It 11.26

Write y=−2x2+8x7y=−2x2+8x7 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
x=ay2+by+cx=ay2+by+c
Standard form
x=a(yk)2+hx=a(yk)2+h
Orientation a>0a>0 right; a<0a<0 left a>0a>0 right; a<0a<0 left
Axis of symmetry y=b2ay=b2a y=ky=k
Vertex Substitute y=b2ay=b2a and
solve for x.
(h,k)(h,k)
y-intercepts Let x=0x=0 Let x=0x=0
x-intercept Let y=0y=0 Let y=0y=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.

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.

How To

Graph horizontal parabolas (x=ay2+by+corx=a(yk)2+h)(x=ay2+by+corx=a(yk)2+h) using properties.

  1. Step 1. Determine whether the parabola opens to the left or to the right.
  2. Step 2. Find the axis of symmetry.
  3. Step 3. Find the vertex.
  4. Step 4. Find the x-intercept. Find the point symmetric to the x-intercept across the axis of symmetry.
  5. Step 5. Find the y-intercepts.
  6. Step 6. Graph the parabola.

Example 11.14

Graph x=2y2x=2y2 by using properties.

Try It 11.27

Graph x=y2x=y2 by using properties.

Try It 11.28

Graph x=y2x=y2 by using properties.

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

Example 11.15

Graph x=y2+2y+8x=y2+2y+8 by using properties.

Try It 11.29

Graph x=y24y+12x=y24y+12 by using properties.

Try It 11.30

Graph x=y2+2y3x=y2+2y3 by using properties.

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(yk)2+h.x=a(yk)2+h. We will use this procedure in the next example.

Example 11.16

Graph x=2(y2)2+1x=2(y2)2+1 using properties.

Try It 11.31

Graph x=3(y1)2+2x=3(y1)2+2 using properties.

Try It 11.32

Graph x=2(y3)2+2x=2(y3)2+2 using properties.

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

Example 11.17

Graph x=−4(y+1)2+4x=−4(y+1)2+4 using properties.

Try It 11.33

Graph x=−4(y+2)2+4x=−4(y+2)2+4 using properties.

Try It 11.34

Graph x=−2(y+3)2+2x=−2(y+3)2+2 using properties.

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

Example 11.18

Write x=2y2+12y+17x=2y2+12y+17 in standard form and then use the properties of the standard form to graph the equation.

Try It 11.35

Write x=3y2+6y+7x=3y2+6y+7 in standard form and use properties of the standard form to graph the equation.

Try It 11.36

Write x=−4y216y12x=−4y216y12 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.

Example 11.19

Find the equation 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 10 feet high and 20 feet wide at the base.
Try It 11.37

Find the equation 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 20 feet high and 40 feet wide at the base.
Try It 11.38

Find the equation 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 5 feet high and 10 feet wide at the base.

Media Access Additional Online Resources

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

Section 11.2 Exercises

Practice Makes Perfect

Graph Vertical Parabolas

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

53.

y=x2+4x3y=x2+4x3

54.

y=x2+8x15y=x2+8x15

55.

y=6x2+2x1y=6x2+2x1

56.

y=8x210x+3y=8x210x+3

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

57.

y=x2+2x4y=x2+2x4

58.

y=2x2+4x+6y=2x2+4x+6

59.

y=−2x24x5y=−2x24x5

60.

y=3x212x+7y=3x212x+7

Graph Horizontal Parabolas

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

61.

x=−2y2x=−2y2

62.

x=3y2x=3y2

63.

x=4y2x=4y2

64.

x=−4y2x=−4y2

65.

x=y22y+3x=y22y+3

66.

x=y24y+5x=y24y+5

67.

x=y2+6y+8x=y2+6y+8

68.

x=y24y12x=y24y12

69.

x=(y2)2+3x=(y2)2+3

70.

x=(y1)2+4x=(y1)2+4

71.

x=(y1)2+2x=(y1)2+2

72.

x=(y4)2+3x=(y4)2+3

73.

x=(y+2)2+1x=(y+2)2+1

74.

x=(y+1)2+2x=(y+1)2+2

75.

x=(y+3)2+2x=(y+3)2+2

76.

x=(y+4)2+3x=(y+4)2+3

77.

x=−3(y2)2+3x=−3(y2)2+3

78.

x=−2(y1)2+2x=−2(y1)2+2

79.

x=4(y+1)24x=4(y+1)24

80.

x=2(y+4)22x=2(y+4)22

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

81.

x=y2+4y5x=y2+4y5

82.

x=y2+2y3x=y2+2y3

83.

x=−2y212y16x=−2y212y16

84.

x=−3y26y5x=−3y26y5

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

85.
This graph shows circle with center (0, 0) and radius 8 units.
86.
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).
87.
This graph shows circle with center (0, 0) and radius 7 units.
88.
This graph shows a parabola opening downwards with vertex (4, 1) and x intercepts (3, 0) and (5, 0).
89.
This graph shows circle with center (2, 3) and radius 3 units.
90.
This graph shows circle with center (negative 5, negative 2) and radius 2 units.

Solve Applications with Parabolas

91.

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).

This graph shows circle with center (negative 5, negative 2) and radius 2 units.
92.

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).

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.
93.

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).

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.
94.

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).

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

95.

In your own words, define a parabola.

96.

Is the parabola y=x2y=x2 a function? Is the parabola x=y2x=y2 a function? Explain why or why not.

97.

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.

98.

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?

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