SOLUTION: Find the equation of the quadratic function with roots -8 and -6, "a" less than zero, and a vertex at (-7, 2).

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Find the equation of the quadratic function with roots -8 and -6, "a" less than zero, and a vertex at (-7, 2).      Log On


   

Question 716254: Find the equation of the quadratic function with roots -8 and -6, "a" less than zero, and a vertex at (-7, 2).
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
When a number is a root of a polynomial function then (x - that number) is a factor of that function. So the quadratic function you are looking for, in factored form, will be:
f%28x%29+=+a%28x-%28-8%29%29%28x-%28-6%29%29
or
f%28x%29+=+a%28x%2B8%29%28x%2B6%29
In general, the "a" can be any non-zero number. But the problem states that "a" should be less than zero (or negative) and that we must have a vertex at (-7, 2). So there will probably be only one value for "a" that will fit. First we will multiply out the factors. Using FOIL on the last two factors:
f%28x%29+=+a%28x%2Ax%2Bx%2A6%2B8%2Ax%2B8%2A6%29
f%28x%29+=+a%28x%5E2%2B6x%2B8x%2B48%29
f%28x%29+=+a%28x%5E2%2B14x%2B48%29
Distributing the "a":
f%28x%29+=+ax%5E2%2B14ax%2B48a%29

Now we set out to figure out what "a" must be. The vertex, (-7, 2), should fit this equation and we can use this to find "a". Substituting -7 for x and 2 for y/f(x) we get:
2+=+a%28-7%29%5E2%2B14a%28-7%29%2B48a%29
Now we solve for "a". We start by simplifying...
2+=+a%2849%29%2B14a%28-7%29%2B48a%29
2+=+49a%2B%28-98a%29%2B48a%29
2+=+-a%29
Dividing by -1:
-2+=+a