DefinitionA parabola is the set of all points in a plane that are the same distance from a fixed line and a fixed point not on the line.

The fixed point is called the focus of a parabola.

The fixed line is called the directrix.

The distance between the vertex and the focus is the focal length of the parabola.

You can find the equation of a vertical parabola with vertex at the origin. If you denote the focus by (0, c), the directrix is the line with equation y = -c.

The equation will be of the form y = ax2. Then a = 14c.

Graph

Key Concept Parabola

TEKS (4)(B) Write the equation of a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening.

TEKS (1)(B) Use a problem–solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem–solving process and the reasonableness of the solution.

Additional TEKS (1)(A), (1)(E)

TEKS FOCUS

Directrix – the fixed line used to define a parabola

Focal length – the distance between the vertex and the focus of a parabola

Focus (plural: foci) of a parabola – the fixed point used to define a parabola

Formulate – create with careful effort and purpose. You can formulate a plan or strategy to solve a problem.

Strategy – a plan or method for solving a problem

Reasonableness – the quality of being within the realm of common sense or sound reasoning. The reasonableness of a solution is whether or not the solution makes sense.

VOCABULARY

Each point of a parabola is equidistant from a point called the focus and a line called the directrix.

ESSENTIAL UNDERSTANDING

5-4 Focus and Directrix of a Parabola

170 Lesson 5-4 Focus and Directrix of a Parabola

x

directrix

Vertical Parabola

axis of symmetry

y

Focus

y

Horizontal Parabola

directrixFocusaxis of symmetry

x

x

y

O

focus(0, c)

(x, y)

(x, �c)directrix y � �c

Problem 1P

Parabolas With Equation y = ax2

A What is an equation of the parabola with vertex at the origin and focus (0, 2)?

The focus is directly above the vertex.

This is a vertical parabola with vertex at the origin.

The focus is (0, c), so c = 2.

y = 14c

x2 = 14(2)

x2 = 18

x2

B What are the focus and directrix of the parabola with equation y = − 1

12x2 ?

This is a vertical parabola with vertex at the origin and a = - 1

12.

a = 14c = - 1

12

4c = -12

c = -3

Since the vertex is at the origin, knowing c, you can conclude that the focus is the point (0, -3) and the directrix is the line with equation y = 3.

y4

2

O 2�2

�2

x

(0, 2)

y

2

4

x1 4

�2

�4

�4 �1

(0, �3)

y � 3

Vertical Parabola Vertex (0, 0) Vertex (h, k)

Equation y = 14c x 2 y = 1

4c(x - h)2 + k

Focus (0, c) (h, k + c)

Directrix y = -c y = k - c

Horizontal Parabola Vertex (0, 0) Vertex (h, k)

Equation x = 14c y 2 x = 1

4c(y - k)2 + h

Focus (c, 0) (h + c, k)

Directrix x = -c x = h - c

Key Concept Transformations of a Parabola

How can you tell if this is a vertical or a horizontal parabola?The focus and the vertex are on the axis of symmetry. They both lie on the y-axis, so the parabola is vertical.

S

What does the sign of a tell you about the graph?Since a is negative, the parabola opens downward.

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Problem 2

Parabolas With Equation x = ay2

A What is an equation of a parabola with vertex at the origin and directrix x = 1.25?

y4

O 42�2

�4

�4x

x � 1.25

c � �1.25

The directrix lies directly to the right of the vertex.

The parabola is horizontal.

The directrix has equation x = -c, so c = -1.25. Thus,

x = 14c y2 Use the equation for a horizontal parabola.

= 14(-1.25) y

2 Substitute -1.25 for c.

= -15 y2 Simplify.

The equation is x = -15y2.

Check for Reasonableness

The graph is reasonable since it opens in the negative direction and a 6 0.

B What are the vertex, focus, and directrix of the parabola with equation x = 0.75y2?

y

2�2x

c �13

( , 0)13

x � �13

This is a horizontal parabola. The vertex is at the origin and a = 0.75. Thus,

a = 14c = 0.75 Substitute 0.75 for a.

4c = 10.75 Solve for c.

c = 13.

Knowing c, you can conclude that the focus is the point 113, 02. The directrix is

the line with equation x = - 13.

How can you tell if this is a vertical or a horizontal parabola?The directrix is parallel to the y-axis, so this is a horizontal parabola.

What does the sign of a tell you about the graph?Since a is positive, the parabola opens to the right.

172 Lesson 5-4 Focus and Directrix of a Parabola

Problem 3

8 ft

Using Parabolas to Solve Problems

Solar Reflector The parabolic solar reflector pictured has a depth of 2 feet at the center. How far from the vertex is the focus? (What is the focal length?)

Graph the parabola in a coordinate system with vertex (0, 0). The vertical parabola has the form y = 1

4c x 2. Substitute either the point (-4, 2) or the point (4, 2).

2 = 14c (4)2 Substitute.

2 = 164c Simplify.

8c = 16 Solve for c.

c = 2

Therefore the focus is at (0, 2), 2 ft from the vertex. The focal length is 2 ft.

TEKS Process Standard (1)(A)

STEM

y4

O 2 4�2�4

�2

�4

x

(�4, 2) (4, 2)

U

Sc

Gt

What is the shape of the solar reflector?A cross section is part of a parabola and is 8 ft across.

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Problem 4

Writing an Equation Given the Focus and Directrix

What is an equation of a parabola with focus (2, −3) and directrix x = −4 ?

Determine a Solution

2(x - x1)2 + (y - y1)2 = 2(x - x2)2 + (y - y2)2 Distance Formula

2 (x - 2)2 + [y - ( -3)]2 = 2[x - (-4)]2 + (y - y)2 Substitute.

2(x - 2)2 + (y + 3)2 = 2(x + 4)2 Simplify.

(x - 2)2 + (y + 3)2 = (x + 4)2 Square each side.

x2 - 4x + 4 + (y + 3)2 = x2 + 8x + 16 Expand in x.

(y + 3)2 - 4x = 8x + 12 Subtract x2 and 4.

(y + 3)2 = 12x + 12 Add 4x to each side.

(y + 3)2 = 12(x + 1) Distributive Property

x = 112(y + 3)2 - 1 Standard Quadratic Form

Evaluate the Reasonableness of the Solution

The equation can be written as x = 14 # 3(y + 3)2 - 1, so c is 3, k = -3, and h is -1.

So, the focus of this parabola is (h + c, k) or (2, -3) and the directrix is x = h - c

or x = -4. The equation is reasonable.

TEKS Process Standard (1)(B)

Analyze the Given InformationYou know the coordinates of the focus and the equation of the directrix. Any point on the parabola is equidistant from the focus and the directrix.

Formulate a PlanBecause you know the distances are equal, use the Distance Formula.

D

2

Why do you substitute y for y2 in the expression on the right?The directrix is x = -4. Any point that lies on that line is of the form(-4, y).

Why don’t you expand in y? The equation of a horizontal parabola contains (y - k)2, so you can use (y + 3)2 for that part of the equation.

174 Lesson 5-4 Focus and Directrix of a Parabola

PRACTICE and APPLICATION EXERCISES

ONLINE

HO

M E W O RK

For additional support whencompleting your homework, go to PearsonTEXAS.com.

Write an equation of a parabola with vertex at the origin and the given focus.

1. focus at (6, 0) 2. focus at (0, -4) 3. focus at (0, 7)

4. focus at (-1, 0) 5. focus at (2, 0) 6. focus at (0, -5)

Identify the vertex, the focus, and the directrix of the parabola with the given equation. Then sketch the graph of the parabola.

7. y = 4x2 8. y = - 18x2

9. x = y2 10. x = 12y2

Write an equation of a parabola with vertex at the origin and the given directrix.

11. directrix x = -3 12. directrix y = 5 13. directrix y = - 1314. directrix x = 9 15. directrix y = 2.8 16. directrix x = -3.75

17. Apply Mathematics (1)(A) A cross section of a flashlight reflector is a parabola. The bulb is located at the focus. Suppose the bulb is located 14 in. from the vertex of the reflector. Model a cross section of the reflector by writing an equation of a parabola that opens upward and has its vertex at the origin. What is an advantage of this parabolic design?

Identify the vertex, the focus, and the directrix of the parabola with the given equation. Then sketch the graph of the parabola.

18. y = x2 + 4x + 3 19. y = x2 - 6x + 11 20. y = x2 + 8x + 13

21. y = x2 - 2x - 4 22. y = x2 - 8x + 17 23. y = 2x2 + 4x - 2

Problem 5

Writing an Equation of a Parabola

Multiple Choice Which is an equation of the parabola with vertex (3, 7) and focus (5, 7)?

x = 14(y - 7)2 + 3 x = 1

8(y - 7)2 + 3

y = 18(x - 7)2 + 3 x = 1

8(y + 7)2 - 3

The focus is to the right of the vertex, so the parabola is horizontal. Also, (h, k) is (3, 7) and (h + c, k) is (5, 7), so c = 2. Substitute all this information into the equation for a horizontal parabola, x = 1

4c(y - k)2 + h, to get x = 18(y - 7)2 + 3.

The correct answer is C.

Scan page for a Virtual Nerd™ tutorial video.

TTaaT

How do you determine which equation to use?Use the focus and the vertex to determine the orientation of the parabola.

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Write an equation of a parabola with the given vertex and focus.

24. vertex (4, 1), focus (6, 1) 25. vertex (0, 3), focus (-8, 3)

26. vertex (-5, 4), focus (-5, 0) 27. vertex (7, 2), focus (7, -2)

28. The center main cable of the extension bridge shown is parabolic. The x-axis represents the bridge roadway and the directrix. The y-axis is the axis of symmetry. Write an equation to model the main cable of the bridge.

Find the equation of the parabola with the given properties.

29. focus (-3, 5) and directrix x = 3

30. focus (4, 5) and directrix y = -3

31. focus (-2, -1) and directrix x = 8

Identify the vertex, the focus, and the directrix of a parabola with each equation. Then sketch a graph of the parabola with the given equation.

32. y2 - 25x = 0 33. x2 = -4y

34. (x - 2)2 = 4y 35. -8x = y2

36. y2 - 6x = 18 37. x2 + 24y - 8x = -16

38. Apply Mathematics (1)(A) In some solar collectors, a mirror with a parabolic cross section is used to concentrate sunlight on a pipe, which is located at the focus of the mirror as shown in the diagram. What is an equation of the parabola that models the cross section of the mirror?

39. Apply Mathematics (1)(A) The equation d = 110 s2 relates the

depth d (in meters) of the ocean to the speed s (in m/s) at which tsunamis travel. What is the graph of the equation?

Use the information in each graph to write the equation for the parabola.

40. 41. 42.

50 ft

6 ft

F

2

y4

2�2

�4

�6

F(�2, 0)

x2

Directrix

y

O

2

�2y � �1

x2

y

3

1

O�2

�2

F Q , 0Rx

14

176 Lesson 5-4 Focus and Directrix of a Parabola

TEXAS Test PracticeT

57. What is the equation of a parabola with vertex at the origin and focus at 10, 522?

A. x = - 110 y2 C. x = - 1

10 x2

B. x = 110y2 D. y = 1

10 x2

58. Use the information in the graph to find the equation for the graph.

F. y2 + 6x = 0 H. x2 + 6y = 0

G. y2 - 6x = 0 J. x2 - 6y = 0

59. Which expression is NOT equivalent to 125x4y213?

A. x 3125xy C. 3225x4y

B. 5x 31xy D. 62625x8y2

y4

O 42�2

�4

�4x

Directrix,x � 3

2

43. Apply Mathematics (1)(A) Broadcasters use a parabolic microphone on football sidelines to pick up field audio for broadcasting purposes. A certain parabolic microphone has a reflector dish with a diameter of 28 inches and a depth of 14 inches. If the receiver of the microphone is located at the focus of the reflector dish, how far from the vertex should the receiver be positioned?

Graph each equation.

44. y2 - 8x = 0 45. y2 - 8y + 8x = -16 46. 2x2 - y + 20x = -53

47. x2 = 12y 48. y = 4(x - 3)2 - 2 49. (y - 2)2 = 4(x + 3)

Write an equation of a parabola with vertex at (1, 1) and the given information.

50. directrix y = - 12 51. directrix x = 32 52. focus at (1, 0)

53. Explain Mathematical Ideas (1)(G) Explain how to find the distance from the focus to the directrix of the parabola x = 2y2.

54. Use a Problem-Solving Model (1)(B) Use the definition of a parabola to show that the parabola with vertex (h, k) and focus (h, k + c) has the equation (x - h)2 = 4c(y - k).

55. a. What part of a parabola is modeled by the function y = 1x?

b. State the domain and range for the function in part (a).

56. Justify Mathematical Arguments (1)(G) If the radius and depth of a satellite dish are equal, prove that the radius is four times the focal length.

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