17.2 Equation of a Parabola

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Explore 1 Identify Points That Are Equidistant From a Point and a Line

Remember that the distance from a point to a line is the length of the perpendicular segment from the point to the line. In the figure, the distance from point A to line ℓ is AB.

You will use the idea of the distance from a point to a line below.

You are given the point R and line ℓ as shown on the graph. Follow these instructions to plot a point.

A Choose a point on line ℓ. Plot point Q at this location.

B Using a straightedge, draw a perpendicular to ℓ that passes through point Q. Label this line m.

C Use the straightedge to draw _ RQ . Then use a compass and straightedge to construct the

perpendicular bisector of _ RQ .

D Plot a point X where the perpendicular bisector intersects the line m.

E Write the approximate coordinates of point X.

F Repeat Steps A–D to plot multiple points. You may want to work together with other students, plotting all of your points on each of your graphs.

Module 17 901 Lesson 2

17.2 Equation of a ParabolaEssential Question: How do you write the equation of a parabola that opens up or down given

its focus and directrix?

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Reflect

The figure shows the construction of point Q, along with several other points constructed using the same method.

1. Explain why point X is equidistant from point R and line ℓ.

2. What do you notice about the points you plotted?

Explore 2 Deriving the Equation of a ParabolaA parabola is the set of all points P in a plane that are equidistant from a given point, called the focus, and a given line, called the directrix. The vertex of a parabola is the midpoint of the segment, perpendicular to the directrix, that connects the focus and the directrix.

For the parabola shown, the vertex is at the origin.

To derive the general equation of a parabola, you can use the definition of a parabola, the distance formula, and the idea that the distance from a point to a line is the length of the perpendicular segment from the point to the line.

A Let the focus of the parabola be F (0, p) and let the directrix be the line y = -p. Let P be a point on the parabola with coordinates (x, y) .

B Let Q be the point of intersection of the perpendicular from P and

the directrix. Then the coordinates of Q are .



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C By the definition of a parabola, FP = QP.

By the distance formula, FP = √ ――――――― (x - 0) 2 + (y - p) 2 = √ ――――― x 2 + (y - p) 2 and

QP = √ ―――――――― (x - x) 2 + (y - (-p) ) 2 = √ ――――― 0 + (y + p) 2 = ⎜y + p⎟ .

Set FP equal to QP. = ⎜y + p⎟

Square both sides. x 2 + = ⎜y + p⎟ 2

Expand the squared terms. x 2 + y 2 - + p 2 = y 2 + + p 2

Subtract y 2 and p 2 from both sides. =

Add 2py to both sides. x 2 =

Solve for y. 1 _

x 2 = y

Reflect

3. Explain how the value of p determines whether the parabola opens up or down.

4. Explain why the origin (0, 0) is always a point on a parabola with focus F (0, p) and directrix y = -p.

Explain 1 Writing an Equation of a Parabola with Vertex at the Origin

You can use the focus and directrix of a parabola to write an equation of the parabola with vertex at the origin.

Example 1 Write the equation of the parabola with the given focus and directrix. Then graph the parabola.

A focus: (0, 5) ; directrix : y = -5

• The focus of the parabola is (0, p) , so p = 5.

The general equation of a parabola is y = 1 __ 4p x 2 .

So, the equation of this parabola is y = 1 ___ 4 (5) x 2 or y = 1 __ 20 x 2 .

• To graph the parabola, complete the table of values. Then plot points and draw the curve.

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-10 5

-5 1.25

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5 1.25

10 5


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B Write the equation of the parabola with focus (0, -4) and directrix y = 4.

• The focus of the parabola is (0, p) , so p = .

The general equation of a parabola is y = 1 __ 4p x 2 .

So, the equation of this parabola is .

• To graph the parabola, complete the table of values. Then plot points and draw the curve.

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Reflect

5. Describe any symmetry in your graph from Example 1B. Why does this make sense based on the parabola’s equation?

Your Turn

Write the equation of the parabola with the given focus and directrix. Then graph the parabola.

6. Parabola with focus (0, -1) and directrix y = 1.

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Explain 2 Writing the Equation of a Parabola with Vertex Not at the Origin

The vertex of a parabola may not be at the origin. But given the focus and directrix of a parabola, you can find the coordinates of the vertex and write an equation of the parabola.

A parabola with vertex (h, k) that opens up or down has equation (x - h) 2 = 4p (y - k) where ⎜p⎟ is the distance from the vertex to the focus.

Example 2 Write the equation of the parabola with the given focus and directrix. Then graph the parabola.

A focus: (3, 5) , directrix: y = -3

• Draw the focus and the directrix on the graph.

• Draw a segment perpendicular to the directrix from the focus.

• Find the midpoint of the segment. The segment is 8 units long, so the vertex is 4 units below the focus. Then p = 4, and the coordinates of the vertex are (3, 1) .

• Since the formula for a parabola is (x - h) 2 = 4p (y - k) , the equation of the parabola is (x - 3) 2 = 16 (y - 1) .

• To graph the parabola, complete the table of values. Round to the nearest tenth if necessary. Then plot the points and draw the curve.

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1 1.3

2 1.1

3 1

4 1.1

6 1.6

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B focus: (-1, 4) ; directrix: y = 6

• Draw the focus and the directrix on the graph.

• Draw a segment perpendicular to the directrix through the focus.

• Find the midpoint of the segment. The segment is units long,

so the vertex is units above/below the focus. The coordinates

of the vertex are . The parabola opens down

so p = .

• Since the formula for a parabola is (x - h) 2 = 4p (y - k) , the equation of the parabola is

(x - ) 2 = (y - ) .

• To graph the parabola, complete the table of values. Round to the nearest tenth if necessary. Then plot the points and draw the curve.

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Reflect

7. Discussion Without calculating, how can you determine by considering the focus and directrix of a parabola whether the parabola opens up or down?


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Your Turn


8. focus: (3, 2) , directrix: y = -4

9. focus: (2, 1) , directrix: y = 9

Elaborate

10. What does the sign of p in the equation of a parabola tell you about the parabola?

11. Explain how to choose appropriate x-values when completing a table of values given the equation of a parabola.

12. Essential Question Check-In How is an equation for the general parabola related to the equation for a parabola with a vertex of (0, 0) ?



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• Online Homework• Hints and Help• Extra Practice


1. focus: (0, 2) ; directrix: y = -2

2. focus: (0, -3) ; directrix: y = 3

3. focus: (0, -5) ; directrix: y = 5


Find the directrix and focus of a parabola with the given equation.

5. y = - 1 _ 24 x 2

Evaluate: Homework and PracticeEvaluate: Homework and Practice



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6. y = 2x 2

7. y = - 1 __ 2 x 2

8. y = 1 __ 40 x 2


9. focus: (2, 3) ; directrix: y = 7

10. focus: (-3, 2) ; directrix: y = -4


12. focus: (-3, 8) ; directrix: y = 9



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Find the focus and directrix of a parabola with the given equation.

13. (x + 2) 2 = 16 (y + 1)

14. (x - 4) 2 = 40 (y - 4)

15. - (x + 5) 2 = y + 3.25

16. (x - 3) 2 = -24 (y + 4)

17. Make a Conjecture Complete the table by writing the equation of each parabola. Then use a calculator to graph the equations in the same window to help you make a conjecture: What happens to the graph of a parabola as the focus and directrix move apart?

Focus (0, 1) (0, 2) (0, 3) (0, 4)

Directrix y = -1 y = -2 y = -3 y = -4

Equation



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18. Find the length of the line segment that is parallel to the directrix of a parabola, that passes through the focus, and that has endpoints on the parabola. Explain your reasoning.

19. Represent Real-World Problems The light from the lamp shown has parabolic cross sections. Write an equation for a cross section of the lamp if the bulb is 6 inches from the vertex and the vertex is placed at the origin. (Hint: The bulb of the lamp is the focus of the parabola.)

20. At a bungee-jumping contest, Gavin makes a jump that can be modeled by the equation (x - 6) 2 = 12 (y - 4) with dimensions in feet.

a. Which point on the path identifies the lowest point that Gavin reached? What are the coordinates of this point? How close to the ground was he?

b. Analyze Relationships Nicole makes a similar jump that can be modeled by the equation (x - 2) 2 = 8 (y - 8.5) . How close to the ground did Nicole get? Did Nicole get closer to the ground than Gavin? If so, by how much?

H.O.T. Focus on Higher Order Thinking

21. Critical Thinking Some parabolas open to the left or right rather than up or down. For such parabolas, if p > 0, the parabola opens to the right. If p < 0, the parabola opens to the left. What is the value of p for the parabola shown? Explain your reasoning.



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22. Critical Thinking Which equation represents the parabola shown?

A. (x - 2) 2 = -16 (y - 2)

B. (y - 2) 2 = -16 (x - 2)

C. (x - 2) 2 = 16 (y - 2)

D. (y - 2) 2 = 16 (x - 2)

23. Amber and James are lying on the ground, each tossing a ball into the air, and catching it. The paths of the balls can be represented by the following equations, with the x-values representing the horizontal distances traveled thrown and the y-values representing the heights of the balls, both measured in feet.

Amber: -20 (x - 15) 2 = y - 11.25 James: -24 (x - 12) 2 = y - 6

a. What are the bounds of the equations for the physical situation? Explain.

b. Whose ball went higher? Which traveled the farthest horizontal distance? Justify your answers.

Lesson Performance Task

Suppose a park wants to build a new half-pipe structure for the skateboarders. To make sure all parts of the ramp are supported, the inside of the ramp needs to be equidistant from a focus point, which is at a height of 3 feet. Find the equation of the half-pipe and sketch it, along with the focus point and the directrix. Would the ramp get steeper or flatten out if the focus point height were at a height of 4 feet? Explain.


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17.2 Equation of a Parabola