Question

which of the following could be a graph of the function below?
$f(x) = (x + 2)(x - 6)(x^2 + 1)$
four graphs labeled a, b, c, d are shown, with a and b having upward - opening ends, c and d having one end up and one end down. a has a root at x = - 2 and x = 6 (approximate), b has a different curve shape, c has a root at x = 6 (approximate) and starts from the bottom left, d has a different curve shape

Explanation:

Step1: Find x-intercepts

Set $f(x)=0$, solve $(x+2)(x-6)(x^2+1)=0$.
$x+2=0 \implies x=-2$; $x-6=0 \implies x=6$; $x^2+1=0$ has no real roots. So x-intercepts are $x=-2$ and $x=6$.

Step2: Determine end behavior

Expand leading terms: $x \cdot x \cdot x^2 = x^4$. For even-degree polynomial with positive leading coefficient: as $x\to+\infty$, $f(x)\to+\infty$; as $x\to-\infty$, $f(x)\to+\infty$.

Step3: Test sign between intercepts

Test $x=0$: $f(0)=(0+2)(0-6)(0+1)=2\times(-6)\times1=-12<0$. So the graph is below x-axis between $x=-2$ and $x=6$.

Step4: Match with options

Only option A has x-intercepts at $x=-2$ and $x=6$, correct end behavior, and is below x-axis between the intercepts.

Answer:

A. <The graph with x-intercepts at x=-2 and x=6, opening upwards at both ends, and dipping below the x-axis between the intercepts>