Write an equation of the parabola that passes through the point (−6,64) and has x-intercepts −4 and 58. Then find the average rate of change from x=−4 to x=6.

Question

Algebra question, please help

Answer 1

Hi Bethany

If (-6,64) the coordinates that your parabola passes through?

Then your parabola is

f(x) = x²/2 - 27x - 116

Based on your x intercepts

Your Parabola looks like

f(x) = a(x + 4)(x - 58)

since it passes through (-6,64) we can use that to find a

64 = a(-6+4)(-6-58)

64 =a(-2)(-64)

64 =a(128)

64/128 =a

1/2 =a

Multiplying gives the standard form

f(x) = x²/2 -27x - 116 where

a = 1, b = -27 and c = -116

The x = h coordinate of the vertex is -b/2a = -(-27)/2(1/2) = 27/1 = 27

Plugging in 27 for x to get f(x) =y = k coordinate of the vertex

729/2 -729-116 = -480.5

vertex of your parabola is (27, -480.5)

Also vertex form of your parabola looks

f(x) = 1/2(x - 27)² - 961/2

You can plot this at desmos.com to see what it looks like.

f(x) = x²/2 - 27x - 116 passes through (-6, 64), (-4,0) and (58,0)

I do hope this helps