Download presentation

Presentation is loading. Please wait.

Published byAlan Stevens Modified over 4 years ago

1
**Sullivan Algebra and Trigonometry: Section 10.2 The Parabola**

Objectives of this Section Find the Equation of a Parabola Graph Parabolas Discuss the Equation of a Parabola Work With Parabolas with Vertex at (h,k)

2
A parabola is defined as the collection of all points P in the plane that are the same distance from a fixed point F as they are from a fixed line D. The point F is called the focus of the parabola, and the line D is its directrix. As a result, a parabola is the set of points P for which: d(F,P) = d(P,D) The equation of a parabola with vertex at (0,0), focus at (a,0), and directrix x = -a is y2 = 4ax

3
D: x = -a F = (a,0) d(F,P) P d(P,D) d(F,P) = d(P,D)

4
**Find the equation of a parabola with vertex at (0,0) and focus at (4,0). Graph the equation.**

The distance from the vertex to the focus is a = 4. So the equation of the parabola is y2 = 16x x = -4 (4,0) (1,4)

5
**Discuss the equation: y2 = 10x**

The equation is of the form y2 = 4ax, where a = 10, so a = 5/2. So, the graph of the equation is a parabola with vertex (0,0), a focus at the point (5/2, 0) and directrix x = -5/2

6
**Equations of a Parabola: Vertex at (0,0); Focus on Axis**

Vertex Focus Directrix Equation Description (0,0) (a,0) x = -a y2 = 4ax Parabola, symmetric on x axis, opens right (0,0) (-a,0) x = a y2 = -4ax Parabola, symmetric on x axis, opens left (0,0) (0,a) y = -a x2 = 4ay Parabola, symmetric on y axis, opens up (0,0) (0,-a) y = a x2 = -4ay Parabola, symmetric on y axis, opens down

7
**Find the equation of a parabola with focus (0, -3) and directrix the line y = 3.**

This parabola will have a vertex at (0,0), since that point is the midpoint between the directrix and the focus. Since the focus is on the negative y axis with a = 3, the equation of the parabola is x2 = -4(3)y or x2 = -12y

8
**Parabolas With Vertex at (h,k); Axis of Symmetry Parallel to a Coordinate Axis**

Vertex Focus Directrix Equation (h,k) (h+a, k) x = h - a (y - k)2 = 4a(x - h) (h,k) (h-a, k) x = h + a (y - k)2 = -4a(x - h) (h,k) (h, k+a) x = k - a (x - h)2 = 4a(y- k) (h,k) (h, k - a) x = k + a (x - h)2 = -4a(y- k)

9
**Find the equation of a parabola with vertex at (-2, 3) and focus at (0, 3). Graph the equation.**

The vertex and focus both lie on the horizontal line y = 3 (the axis of symmetry). This distance from the vertex to the focus is a = 2. Since the focus lies to the right of the vertex, the parabola opens to the right. The equation has the form: (y - k)2 = 4a(x - h) where (h,k) = (-2, 3) and a = 2 (y - 3)2 = 4(2)(x - (-2)) (y - 3)2 = 8(x + 2)

10
**(y - 3)2 = 8(x + 2) (0, 7) D: x = -4 F = (0,3) Axis of Symmetry: y = 3**

(0, -1) (y - 3)2 = 8(x + 2)

11
**Find the vertex and focus of the following parabola:**

x2 + 8x = 4y - 8 First, use the method of completing the square involving the variable x to rewrite the parabola. x2 + 8x + ____ = 4y ____ 16 16 (x + 4)(x + 4) = 4y + 8 (x + 4)2 = 4(y + 2) The equation is of the form (x - h)2 = 4a(y- k) Vertex: (h,k) = (-4, -2); Focus: (h, k + a) = (-4, -1)

Similar presentations

© 2020 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google