Determine the equation of each of the following parabolas

Step-by-step explanation:

For graphs where the zeros (x-intercepts) are known, use the factored form of a parabola:

y = a (x − b) (x − c)

where (b, 0) and (c, 0) are the zeros.  Plug in a third point to find the value of a.

For graphs where the vertex is known, use the vertex form of a parabola:

y = a (x − h)² + k

where (h, k) is the vertex.  Plug in a second point to find the value of a.

I'll do the first three problems as examples.

a) The vertex is (0, 5), and one of the zeros is (4, 0).  Since the vertex is halfway between the zeros, we also know the other zero is (-4, 0).  So we can use either form.

Using factored form:

y = a (x − 4) (x − (-4))

y = a (x − 4) (x + 4)

Plug in the vertex to find the value of a:

5 = a (0 − 4) (0 + 4)

5 = -16a

a = -5/16

Therefore, the equation is:

y = -5/16 (x − 4) (x + 4)

b) The vertex is (0, 0).  It is also the only root.  So again, we can use either form.

Using vertex form:

y = a (x − 0)² + 0

y = ax²

Plugging in the second point:

9 = a (-3)²

9 = 9a

a = 1

Therefore, the equation is:

y = x²

c) This time, we don't know the coordinates of the vertex, but we do know the roots are (-7, 0) and (0, 0).  Using factored form:

y = a (x − 0) (x − (-7))

y = a x (x + 7)

Plugging in the third point:

4 = a (4) (4 + 7)

4 = 44a

a = 1/11

Therefore, the equation is:

y = 1/11 x (x + 7)