Chapter 12
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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

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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.


Parabola

Parabola


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

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

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! Example 1

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


Parabolas2

Parabolas

  • 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.


Standard form for the equation

Standard form for the equation


Example 2a writing equations of parabolas

Example 2A: Writing Equations of Parabolas

  • Write the equation in standard form for the parabola.


Solution2

solution

  • 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

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


Example

Example

  • 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

  • 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! Example 2a

  • Write the equation in standard form for the parabola.

  • vertex (0, 0), directrixx = 1.25


Parabolas3

Parabolas

  • 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


Example 3 graphing parabolas

Example 3: Graphing Parabolas

  • 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

  • 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


Check it out example

Check it-out Example

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


Parabolas4

Parabolas

  • 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 = 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

  • Do problems 2-8 in your book page 849


Homework

Homework

  • Do problems 14-21 in your book page 849


Closure

Closure

  • Today we learned about parabolas

  • Next class we are going to learn Identifying conic sections


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