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Chapter 12 . 12-5 Parabolas. Objectives . Write the standard equation of a parabola and its axis of symmetry. Graph a parabola and identify its focus, directrix , and axis of symmetry. parabolas.

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Chapter 12

Chapter 12

12-5 Parabolas


Objectives
Objectives

Write the standard equation of a parabola and its axis of symmetry.

Graph a parabola and identify its focus, directrix, and axis of symmetry.


Parabolas
parabolas

  • 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.


Parabolas1
Parabolas

  • 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.



Example 1 using the distance formula to write the equation of a parabola
Example 1: Using the Distance Formula to Write the Equation of a Parabola

  • Use the Distance Formula to find the equation of a parabola with focus F(2, 4) and directrixy = –4.

  • Sol.

  • PF = PD

  • Distance Formula.


Solution
solution of a Parabola

Substitute (2, 4) for (x1, y1) and (x, –4) for (x2, y2).

Simplify.

(x – 2)2 + (y – 4)2 = (y + 4)2

Square both sides.

(x – 2)2 + y2 – 8y + 16 = y2 + 8y + 16


Solution1
solution of a Parabola

Subtract y2 and 16 from both sides.

  • (x – 2)2 – 8y = 8y

(x – 2)2 = 16y

Add 8y to both sides.

Solve for y


Check it out example 1
Check It Out! of a Parabola Example 1

  • Use the Distance Formula to find the equation of a parabola with focus F(0, 4) and directrixy = –4.


Parabolas2
Parabolas of a Parabola

  • 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.



Example 2a writing equations of parabolas
Example 2A: Writing Equations of Parabolas of a Parabola

  • Write the equation in standard form for the parabola.


Solution2
solution of a Parabola

  • Step 1 Because the axis of symmetry is vertical and the parabola opens downward, the equation is in the form

  • Step 2 The distance from the focus (0, –5) to the vertex (0, 0), is 5, so p = –5 and 4p = –20.

y = 1/4p x2 with p < 0.


Solution3
solution of a Parabola

  • Step 3 The equation of the parabola is y = – 1/20 x2


Example
Example of a Parabola

  • Write the equation in standard form for the parabola.

  • vertex (0, 0), directrixx = –6

  • Solution:

  • 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.


Solution4
solution of a Parabola

  • Step 2 Because the directrix is x = –6, p = 6 and 4p = 24.

  • Step 3 The equation of the parabola is

x = 1/24 y2


Check it out example 2a
Check It Out! of a Parabola Example 2a

  • Write the equation in standard form for the parabola.

  • vertex (0, 0), directrixx = 1.25


Parabolas3
Parabolas of a Parabola

  • 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.


Standard form
Standard form of a Parabola


Example 3 graphing parabolas
Example 3: Graphing Parabolas of a Parabola

  • Find the vertex, value of p, axis of symmetry, focus, and directrix of the parabola y + 3= 1/8 (x –2)2.Then graph.

  • Solution:

  • Step 1 The vertex is (2,–3).

  • Step 2 1/4p=1/8 , so 4p = 8 and p = 2.


Solution5
solution of a Parabola

  • Step 3 The graph has a vertical axis of symmetry, with equation x = 2, and opens upward.

  • 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.


Solution6
solution of a Parabola


Check it out example
Check it-out Example of a Parabola

  • Find the vertex, value of p, axis of symmetry, focus, and directrix of the parabola. Then graph.


Parabolas4
Parabolas of a Parabola

  • 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.


Example 4 using the equation of a parabola

x of a Parabola= y2,

x = y2.

1

1

132

4p

The equation for the cross section is in the form

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.

Example 4: Using the Equation of a Parabola

  • The cross section of a larger parabolic microphone can be modeled by the equation What is the length of

  • the feedhorn?

  • Solution:


Student guided practice
Student Guided Practice of a Parabola

  • Do problems 2-8 in your book page 849


Homework
Homework of a Parabola

  • Do problems 14-21 in your book page 849


Closure
Closure of a Parabola

  • Today we learned about parabolas

  • Next class we are going to learn Identifying conic sections


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