Question

Write the equation of the quadratic function with roots -9 and and -3 and a vertex at (-6, -1).

Answer 1

The equation of the quadratic function is:  $y=\frac{1}{9}x^2+\frac{4}{3}x+3$

Explanation

The two roots of the quadratic function are given as  -9 and -3

So, the associative factors for those two roots will be $(x+9)$ and $(x+3)$

Thus, the quadratic function will be:  $y=a(x+9)(x+3)........................(1)$

Now the vertex is at (-6, -1). As the vertex lies on the graph of this quadratic function, so that vertex point will satisfy equation (1).

So, plugging x= -6 and y = -1 into the equation (1)..........

$-1=a(-6+9)(-6+3)\\ \\ -1=a(3)(-3)\\ \\ -1=-9a\\ \\ a= \frac{1}{9}$

So, the quadratic function will be.....

$y=\frac{1}{9}(x+9)(x+3)\\ \\ y=\frac{1}{9}(x^2+12x+27)\\ \\ y=\frac{1}{9}x^2+\frac{4}{3}x+3$