This document provides an overview of parabolas including their key characteristics and equations. It defines a parabola as the set of all points that are an equal distance from both a fixed point called the focus and a fixed line called the directrix. The standard equation of a parabola is provided and examples are given of writing the equation of a parabola given its focus and directrix. The document also discusses graphing parabolas by identifying the vertex, axis of symmetry, focus, and directrix. Real-world applications of parabolas to reflect light and sound are briefly described.

1. UNIT 13.2 PARABOLAS

2. Warm Up
132. from (0, 2) to (12, 7)
Find each distance.
3. from the line y = –6 to (12, 7) 13
1. Given , solve for p when c =

3. Write the standard equation of a
parabola and its axis of symmetry.
Graph a parabola and identify its focus,
directrix, and axis of symmetry.
Objectives

4. focus of a parabola
directrix
Vocabulary

5. In Chapter 5, you learned
that the graph of a
quadratic function is a
parabola. Because a
parabola is a conic section,
it can also be defined in
terms of distance.

6. A parabola is the set of all
points P(x, y) in a plane that
are an equal distance from
both a fixed point, the
focus, and a fixed line, the
directrix. A parabola has a
axis of symmetry
perpendicular to its directrix
and that passes through its
vertex. The vertex of a
parabola is the midpoint of
the perpendicular segment
connecting the focus and the
directrix.

7. The distance from a point to a line is defined as
the length of the line segment from the point
perpendicular to the line.
Remember!

8. Use the Distance Formula to find the equation of a
parabola with focus F(2, 4) and directrix y = –4.
Example 1: Using the Distance Formula to Write the
Equation of a Parabola
Definition of a parabola.PF = PD
Substitute (2, 4)
for (x1, y1) and
(x, –4) for (x2, y2).
Distance
Formula.

9. Example 1 Continued
(x – 2)
2
+ (y – 4)
2
= (y + 4)
2 Square both sides.
Expand.
Subtract y
2
and 16
from both sides.
(x – 2)
2
+ y
2
– 8y + 16 = y
2
+ 8y + 16
(x – 2)
2
– 8y = 8y
(x – 2)
2
= 16y
Add 8y to both
sides.
Solve for y.
Simplify.

10. Use the Distance Formula to find the equation of a
parabola with focus F(0, 4) and directrix y = –4.
Definition of a parabola.PF = PD
Substitute (0, 4)
for (x1, y1) and
(x, –4) for (x2, y2).
Distance Formula
Check It Out! Example 1

11. x
2
+ (y – 4)
2
= (y + 4)
2
Square both sides.
Expand.
Subtract y
2
and 16
from both sides.
x2
+ y2
– 8y + 16 = y2
+ 8y +16
x
2
– 8y = 8y
x
2
= 16y Add 8y to both sides.
Solve for y.
Check It Out! Example 1 Continued
Simplify.

12. Previously, you have graphed parabolas with
vertical axes of symmetry that open upward or
downward. Parabolas may also have horizontal
axes of symmetry and may open to the left or
right.
The equations of parabolas use the parameter p.
The |p| gives the distance from the vertex to
both the focus and the directrix.

14. Write the equation in standard form for the
parabola.
Example 2A: Writing Equations of Parabolas
Step 1 Because the axis
of symmetry is vertical
and the parabola opens
downward, the equation
is in the form
y = x
2
with p < 0.
1
4p

15. Example 2A Continued
Step 2 The distance from the focus (0, –5) to
the vertex (0, 0), is 5, so p = –5 and 4p = –20.
Step 3 The equation of the parabola is .y = – x
21
20
Check
Use your graphing
calculator. The
graph of the
equation appears to
match.

16. Example 2B: Writing Equations of Parabolas
vertex (0, 0), directrix x = –6
Write the equation in standard form for the
parabola.
Step 1 Because the directrix is a vertical
line, the equation is in the form . The
vertex is to the right of the directrix, so the
graph will open to the right.

17. Example 2B Continued
Step 2 Because the directrix is x = –6, p = 6
and 4p = 24.
Step 3 The equation of the parabola is .x = y
21
24
Check
Use your graphing
calculator.

18. vertex (0, 0), directrix x = 1.25
Check It Out! Example 2a
Write the equation in standard form for the
parabola.
Step 1 Because the directrix is a vertical line,
the equation is in the form of . The
vertex is to the left of the directrix, so the
graph will open to the left.

19. Step 2 Because the directrix is x = 1.25,
p = –1.25 and 4p = –5.
Check
Check It Out! Example 2a Continued
Step 3 The equation of the parabola is
Use your graphing
calculator.

20. Write the equation in standard form for each
parabola.
vertex (0, 0), focus (0, –7)
Check It Out! Example 2b
Step 1 Because the axis of symmetry is
vertical and the parabola opens downward,
the equation is in the form

21. Step 2 The distance from the focus (0, –7) to
the vertex (0, 0) is 7, so p = –7 and
4p = –28.
Check
Check It Out! Example 2b Continued
Use your graphing
calculator.
Step 3 The equation of the parabola is

22. The vertex of a parabola may not always be the
origin. Adding or subtracting a value from x or y
translates the graph of a parabola. Also notice that
the values of p stretch or compress the graph.

24. Example 3: Graphing Parabolas
Step 1 The vertex is (2, –3).
Find the vertex, value of p, axis of
symmetry, focus, and directrix of the
parabola Then graph.y + 3 = (x – 2)
2
.
1
8
Step 2 , so 4p = 8 and p = 2.1
4p
1
8
=

25. Example 3 Continued
Step 4 The focus is (2, –3 + 2),
or (2, –1).
Step 5 The directrix is a
horizontal line
y = –3 – 2, or
y = –5.
Step 3 The graph has a vertical axis of symmetry,
with equation x = 2, and opens upward.

26. Step 1 The vertex is (1, 3).
Find the vertex, value of p, axis of symmetry,
focus, and directrix of the parabola. Then graph.
Step 2 , so 4p = 12 and p = 3.1
4p
1
12
=
Check It Out! Example 3a

27. Step 4 The focus is (1 + 3, 3),
or (4, 3).
Step 5 The directrix is a
vertical line x = 1 – 3,
or x = –2.
Step 3 The graph has a horizontal axis of
symmetry with equation y = 3, and opens
right.
Check It Out! Example 3a Continued

28. Step 1 The vertex is (8, 4).
Find the vertex, value of p axis of symmetry,
focus, and directrix of the parabola. Then graph.
Check It Out! Example 3b
Step 2 , so 4p = –2 and p = – .1
4p
1
2
= –
1
2

29. Step 3 The graph has a vertical axis of symmetry,
with equation x = 8, and opens downward.
Check It Out! Example 3b Continued
Step 4 The focus is
or (8, 3.5).
Step 5 The directrix is a
horizontal line
or y = 4.5.

30. Light or sound waves collected
by a parabola will be reflected
by the curve through the focus
of the parabola, as shown in
the figure. Waves emitted
from the focus will be reflected
out parallel to the axis of
symmetry of a parabola. This
property is used in
communications technology.

31. The cross section of a larger parabolic
microphone can be modeled by the
equation What is the length of
the feedhorn?
Example 4: Using the Equation of a Parabola
x = y
2
.
1
132
The equation for the cross section is in the form
x = y
2
,
1
4p
so 4p = 132 and p = 33. The focus
should be 33 inches from the vertex of the cross
section. Therefore, the feedhorn should be 33
inches long.

32. Check It Out! Example 4
Find the length of the feedhorn for a microphone
with a cross section equation x = y
2
.
1
44
The equation for the cross section is in the form
x = y2
,
1
4p
so 4p = 44 and p = 11. The focus
should be 11 inches from the vertex of the cross
section. Therefore, the feedhorn should be 11
inches long.

33. Lesson Quiz
1. Write an equation for the parabola with focus
F(0, 0) and directrix y = 1.
2. Find the vertex, value of p, axis of symmetry, focus,
and directrix of the parabola y – 2 = (x – 4)2
,
1
12
then graph.
vertex: (4, 2); focus:
(4,5); directrix: y = –1;
p = 3; axis of
symmetry: x = 4

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