Vertex Form Question?

Question

Given part of an equation of a parabola in vertex form as y = a(x - 8)2 - 2. If this graph passes through the point (4, 6), what is the value of ‘c’ value when the equation is written in standard form?

Answer 1

y = a(x - 8)² - 2

PART 1:

First we need to figure out what value of a will make this pass through the point (4, 6). So plug in x = 4 and y = 6 and solve for a.

Substituting in the point (4, 6):
6 = a(4 - 8)² - 2

Simplify:
6 = a(-4)² - 2
6 = a(16) - 2
6 = 16a - 2

Add 2 to both sides:
6 + 2 = 16a
8 = 16a

Divide both sides by 16:
8/16 = a
1/2 = a
a = 1/2

PART 2:

Next, expand the equation out and see what the 'c' value is.

y = ½(x - 8)² - 2

Start by expanding the square with FOIL:

y = ½(x - 8)(x - 8) - 2
y = ½(x² - 16x + 64) - 2

Distribute the ½ through the parentheses:
y = ½x² - 8x + 32 - 2

Combine the last terms to get it in standard form:
y = ½x² - 8x + 30

a = ½
b = -8
c = 30

There you go!

Answer 2

I think the approach here is to use the point
(4,6) to calculate the value of "a":
Substituting,
6 = a (4-8)^2 -2,
so then it must be a = 1/2.

Then, you can rearrange the equation in
y = ax^2 + bx + c form as follows:

y = 1/2 (x-8)^2 - 2
y = 1/2 (x^2 - 16x + 64) - 2
y = 1/2 x^2 - 8 x + 30 after expanding.

Answer 3

y = a(x - 8)² - 2
6 = (a) 16 - 2
16a = 8
a = 1 / 2
y = (1/2) (x - 8)² - 2
y = (1/2) (x² - 16x + 64) - 2
y = (1/2) x² - 8x + 30
c = 30