You are given a graph and asked to write the equation of the quadratic function.

Question

The graph shows a vertex = (-1,-8) and the zeros at -3 and 1. The directions say to find the sum of the roots, and the product of the roots. Then insert the answers into the general formula of f(x)=a[x^2-(sum of roots)x + (products of roots)].When I did this, the answer for a was 0. Then I inserted this 0 into the formula again y=0(x^2 + 2x - 3). I don't know what to do with this. Can you show me what is wrong and how to fix it?

Answer 1

Christine - Just to clarify that last step in Philip's explanation, imagine for a moment that you had three different parabolas and that each one of them hit the axis at those same two zeroes, but one of the parabolas was more stretched out and the other was less stretched out than your original.  

 

If you calculated the three formulas, what would be the difference between them?

 

That difference, the "stretch factor", is the "a" that Philip solved for in his last step.  

 

 

Answer 2

The roots are at x = 3 and -1, so the equation is f(x) =  a(x+3)(x-1) = a(x² + 4x - 3:)

• Sum of roots = 3 + (-1) = 2
• Product of roots = (3)(-1) = -3

 

f(x) = a{x² - (sum of roots)x + (product of roots)} =  a{x² - 4x - 3}

 

To find a, plug in the x and y coordinates of the vertex, (-1,-8), and solve for a:

 

-8 = a{1 + 4 - 3)

-8 = 2a

-4 = a

 

f(x) = (-2)(x+3)(x-1) = -2x² + 8x + 6