Question

write an equation for the quadratic function $f$ whose graph passes through the points $(-6, 0), (-7, 0),$ and $(-8, -6).$ $f(x)=$

Explanation:

Step1: Use factored quadratic form

Since the graph has x-intercepts at $(-6, 0)$ and $(-7, 0)$, the quadratic can be written as:
$f(x) = a(x+6)(x+7)$

Step2: Solve for coefficient $a$

Substitute the point $(-8, -6)$ into the equation:
$-6 = a(-8+6)(-8+7)$
$-6 = a(-2)(-1)$
$-6 = 2a$
$a = \frac{-6}{2} = -3$

Step3: Expand to standard form

Substitute $a=-3$ back and expand:
$f(x) = -3(x+6)(x+7)$
$f(x) = -3(x^2 + 13x + 42)$
$f(x) = -3x^2 - 39x - 126$

Answer:

$f(x) = -3x^2 - 39x - 126$