Question

find equation of parabola for these four problems and show work. 1. vertex (3,-2) passes through 2,0) 2. vertex (2,5) , focus (2,3) 3.(-2,-4) and directrix y= -6 4.focus (2,-6) and directrix y= 3.

Answer 1

9514 1404 393

Answer:

1. f(x) = 2(x -3)^2 -2
2. f(x) = -1/8(x -2)^2 +5
3. f(x) = 1/x(x +2)^2 -4
4. f(x) = -1/18(x -2)^2 -1.5

Explanation:

The vertex form of the equation of a parabola is ...

f(x) = a(x -h)^2 +k

for vertex (h, k).

The value of 'a' can be chosen to make the curve pass through a given point, or it can be chosen to correspond to some focus or directrix. The vertex is always halfway between the focus and directrix, and the value of 'a' is always 1/(4p) where p is the focus-vertex distance. The parabola always opens toward the focus and away from the directrix, telling you the sign of 'a'.

These facts are used to solve the problems posed.

__

1. f(x) = a(x -3)^2 -2 goes through f(2) = 0. (Equation with given vertex.)

0 = a(2 -3)^2 -2

2 = a . . . . add 2, simplify

f(x) = 2(x -3)^2 -2

__

2. The vertex is (5-3) = 2 units above the focus, so the parabola opens downward (a < 0). 1/(4p) = 1/(4·2) = 1/8, so the equation is ...

f(x) = -1/8(x -2)^2 +5

__

3. The directrix is -4-(-6) = 2 units below the focus, so the parabola opens upward. a = 1/(4p) = 1/(4·2) = 1/8, so the equation is ...

f(x) = 1/8(x +2)^2 -4

__

4. The directrix is 3-(-6) = 9 units above the focus, so p=4.5 and the value of a is -1/4(4.5) = -1/18. The vertex is halfway between the focus and directrix at ...

(2, (-6+3)/2) = (2, -1.5)

The equation of the parabola is ...

f(x) = -1/18(x -2)^2 -1.5

_____

On a graph, the distance from focus to vertex is half the horizontal distance from focus to parabola. That is, each point on the parabola is the same distance from the focus and the directrix.