Given a parabola with a vertex of (6,-18) write the equation of the parabola if you know that it passes through the origin (0,0).

Question

Vertex form = f(x)=a(x-h)^2+k

Vertex = (6,-18)

Point = (0,0)

How do you find the quadratic equation step-by-step using vertex form?

 

Answer 1

The vertex has coordinates (h, k).

 

h = 6

k = -18

 

This means in vertex form, the equation is

 

y = a(x - 6)² - 18

 

 

Since the point passes through the origin, plug in those coordinates into the vertex form to find the value of a.  The coordinate is also an x-intercept because the y value is 0.

 

0 = a(0 - 6)² - 18 

 

0 = a(-6)² - 18

 

0 = 36a - 18

 

18 = 36a

 

1/2 = a

 

 

The equation of the quadratic is

 

y = (1/2)(x - 6)² - 18

 

 

Expand this out.

 

y = (1/2)(x² - 12x + 36) - 18

 

y = (1/2)x² - 6x + 18 - 18

 

y = (1/2)x² - 6x

 

 

Check this on a graphing calculator to see if it crosses the point (0, 0)  and the y value is -18 when the x value is 6.

Answer 2

First, plug in the vertex (6, -18) for (h, k):

 

f(x) = a(x - 6)² - 18

 

Then, plug in (0, 0) for (x, f(x)) since you know that point is on the parabola:

 

0 = a(0 - 6)² - 18

 

Now you can solve for a:

 

18 = a(-6)²

18 = 36a

18/36 = a

1/2 = a

 

Finally, plug the vertex (6, -18) and a = 1/2 into vertex form to get the general equation for the parabola:

 

f(x) = 1/2(x - 6)² - 18

 

If you need to convert this to standard form, then just simplify the right-hand side:

 

f(x) = 1/2(x - 6)(x - 6) - 18

f(x) = 1/2(x² - 6x - 6x + 36) - 18

f(x) = 1/2(x² - 12x + 36) - 18

f(x) = 1/2x² - 6x + 18 - 18

f(x) = 1/2x² - 6x