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

`y = - 2{x}^(2) -28x - 96`

Step-by-step 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`