Question

which equation choice could represent the graph shown below?
answer
$f(x) = (x - 4)(x + 2)(x + 2)$ $f(x) = (x + 4)(x - 2)(x - 2)$
$f(x) = (x - 4)(x - 2)(x - 2)$ $f(x) = (x + 4)(x + 2)(x + 2)$

Explanation:

Step1: Identify x-intercepts

The graph intersects the x-axis at \( x = -4 \) (a single root) and \( x = 2 \) (a repeated root, since the graph touches the axis and turns around there).

Step2: Determine factors from roots

For a root \( x = a \), the corresponding factor is \( (x - a) \). For \( x = -4 \), the factor is \( (x + 4) \). For the repeated root \( x = 2 \), the factor is \( (x - 2) \) with multiplicity 2, so \( (x - 2)^2=(x - 2)(x - 2) \).

Step3: Form the polynomial

Combining these factors, the polynomial is \( f(x)=(x + 4)(x - 2)(x - 2) \).

Answer:

\( f(x) = (x + 4)(x - 2)(x - 2) \) (the second option: \( f(x)=(x + 4)(x - 2)(x - 2) \))