Parabola Equation Problem!

[pog99]

Hey,
I can't seem to figure out how to get the equation for this parabola
The problem reads:
A parabola has intercepts at x=-8, x=4, and y=8. What is the equation of the parabola?

I have figured out that the center of the parabola must be at (-2,0) but i cannot find the k value for the center point (h,k). I also know that the parabola must go though the point (-4,-8) because it is paralel point to (0,-8)

I have been working on this problem for a long time but all the formulas I find arn't applicable to the problem because most explanations call for a minimum or maximum point.

Thanks,
Paul

 

[calchere]

A parabola has intercepts at x=-8, x=4, and y=8. What is the equation of the parabola?

Graphing the parabola helps alot, and comparing it to the parabola x^2 helps too.

The general form for a parabola:

\(\displaystyle \H\
y=ax^2+bx+c\)

Since the the parabola is upside down, the x^2 must be negative.
So, to the general equation:

\(\displaystyle \H\
y= -ax^2+bx+c\)

Since we have an intercept at (0,8), we think that when x is 0, the constant must add 8.
So to the general equation:

\(\displaystyle \H\
y=-ax^2+bx+8\)

Since the center of the parabola is when x= -2, we must shift the graph in the negative direction. To do this, we change bx to -bx.
So, to the general equation:

\(\displaystyle \H\
y=-ax^2-bx+8\)

Notice that this parabola is wider than the parabola x^2. Since it is wider, you divide the x^2 by a number, which means a is a fraction. To find out this number, lets look at the intercept (-8,0). Lets plug that into the equation we have so far.

\(\displaystyle \H\
y= - \frac{1}{a}( x)^2 - bx + 8\)

Let's leave the variable b as one and solve the equation.

\(\displaystyle \H\
- \frac{1}{a}( - 8)^2 - ( - 8) + 8 = 0 \\
- \frac{{64}}{a} + 16 = 0 \\
- \frac{{64}}{a} = - 16 \\
a = 4 \\\)

So, this makes our equation:

\(\displaystyle \H\
y=- \frac{1}{4}( x)^2 - bx + 8\)

This is the equation of the parabola, so b is 1. So the final equation:

\(\displaystyle \H\
y= - (\frac{{x^2 }}{4}) - x + 8\)

 

[pog99]

Thanks so much, I was really stuck but now i get it

----Paul