Question

graph the polynomial function f(x)=x^2(x + 3) using parts (a) through (e). the graph of the function. 2 (type a whole number.) (e) use the information to draw a complete graph of the function. use the graphing tool to graph the function. click to enlarge graph

Explanation:

Step1: Expand the function

$f(x)=x^{2}(x + 3)=x^{3}+3x^{2}$

Step2: Find the x - intercepts

Set $f(x)=0$, then $x^{2}(x + 3)=0$. So $x = 0$ (with multiplicity 2) and $x=-3$.

Step3: Find the y - intercept

Set $x = 0$ in $f(x)$, then $f(0)=0^{2}(0 + 3)=0$.

Step4: Analyze the end - behavior

Since the leading term of $f(x)=x^{3}+3x^{2}$ is $x^{3}$ (odd - degree with positive leading coefficient), as $x\to-\infty$, $y\to-\infty$ and as $x\to+\infty$, $y\to+\infty$.

Step5: Sketch the graph

Plot the intercepts and use the end - behavior to draw a smooth curve.

Answer:

The graph has x - intercepts at $x = 0$ and $x=-3$, y - intercept at $y = 0$, and has the end - behavior of going down to the left and up to the right.