How do I find the equation with a vertex and x intercepts?

Question

How do I find the equation to a question asking:

Use the vertex (-4,8)

and x intercepts at (-6,0) and (-2,0)?

Answer 1

Do you want the equation of the function with these two x intercepts and a vertex (-4,8), if then, that looks like an inverted shape of quadratic function.

We have x intercepts intersecting with the x axis and the vertex (-4,8) seems to be the apex and line of symmetry, then the following should work.

As we have the symmetry x axis, the y = a(x – h)2 + k should be used, where the vertex is (h, k).

y = a(x- (-4))2 + 8

y = a(x + 4)2 + 8

y = a(x + 4)2 + 8

y = a(x2 + 8x +16) +8

You would follow along with the x intercept points to find the value of (a):

(-2,0)

0 = a((-2)2 + 8(-2) +16) +8

0 = a(4 -16 + 16) + 8

0 = a (4) + 8 ==> Then, a = -8/4 = -2

The final equation would be

y = -2(x2 + 8x +16) +8

y = -2x2 -16x -32 + 8

y = -2x2 -16x - 24