Question

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

Answer 1

ANSWER

$y = - 2{x}^{2} -28x - 96$

EXPLANATION

We have that

$x = - 8 \: \: and \: \: x = - 6$

are the roots of the quadratic function.

This implies that

$x + 8 \: \: and \: \: x + 6$

are factors of the quadratic function.

The quadratic function will have an equation of the form:

$y = a(x + 8)(x + 6)$

It was also given that, the vertex of the function is at

$(-7, 2)$

This point must satisfy the equation.

This implies that:

$2= a( - 7 + 8)( - 7+ 6)$

This implies that,

$2=-a$

$a = - 2$

We substitute the value of 'a' to get the equation in factored form as:

$y = - 2(x + 8)(x + 6)$

We expand the parenthesis to write the equation in standard form.

$y = - 2( {x}^{2} + 6x + 8x + 48)$

$y = - 2( {x}^{2} + 14x + 48)$

$y = - 2{x}^{2} -28x - 96$

Or in vertex form, the equation is

$y = - 2 {(x + 7)}^{2} + 2$