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)