A function has a graph that is a parabola with vertex (4,8) and containing the point (1,-7). Write the equation for the function in vertex and standard form.

Answer 1

Given vertex: (4, 8), where h = 4 and k = 8.

Vertex form: y = a(x - h)² + k

Standard form: y = ax² + bx + c

We will use the given point (1, -7) to find the "a" term.

a(1 - 4)² + 8 = -7

a(-3)² + 8 = -7

9a + 8 = -7

9a = -15

a = -15/9

a = -5/3

Vertex form: y = -5/3(x - 4)² + 8

Let's convert from vertex form to standard form.

y = -5/3(x² - 4x - 4x + 16) + 8

y = -5/3(x² - 8x + 16) + 8

y = (-5/3)x² + (40/3)x - 80/3 + 8

y = (-5/3)x² + (40/3)x - 80/3 + (24/3)

Standard form: y = (-5/3)x² + (40/3)x - 56/3

Answer 2

If a parabola has a vertical axis, the standard form of the equation of the parabola is this: (x - h)² = a(y - k), where p≠ 0. The vertex of this parabola is at (h, k).

Plugging in the vertex into the standard equation:

(x-4)² = a(y-8)

Next, plug in the point it contains and solve for a

(1-4)² = a(-7-8) Simplify

(-3)² = -15a

9=-15a Divide both sides by -15

-9/15 = a Reduce

-3/5 = a

Plug in a:

(x-4)² = -3/5(y-8)