Question

determine which of the given graphs is that of the function f(x)=−5(x + 4)(x + 3)(x + 2) (hint: its probably best if you just graph the function on your own and compare.)

Explanation:

Step1: Find the x - intercepts

Set $f(x)=0$, then $-5(x + 4)(x + 3)(x + 2)=0$. Solving for $x$, we get $x=-4,x=-3,x=-2$ since if $ab = 0$, then $a = 0$ or $b=0$.

Step2: Determine the end - behavior

The leading term of the polynomial $f(x)=-5(x + 4)(x + 3)(x + 2)=-5(x^{3}+9x^{2}+26x + 24)=-5x^{3}-45x^{2}-130x - 120$ is $-5x^{3}$. Since the leading coefficient $a=-5<0$ and the degree $n = 3$ (odd), as $x\to-\infty$, $y\to\infty$ and as $x\to\infty$, $y\to-\infty$.

Step3: Check the graphs

Look for the graph that crosses the x - axis at $x=-4,x=-3,x=-2$ and has the correct end - behavior.

Answer:

Based on the above analysis, choose the graph that has x - intercepts at $x=-4,x=-3,x=-2$ and the end - behavior of an odd - degree polynomial with a negative leading coefficient. (Since the actual graphs are not labeled, you would identify the correct one among them using the x - intercepts and end - behavior criteria).