Question

answer parts a. - e. for the function shown below. f(x)=3x^2 + x^3
rises right.
c. the graph of f(x) falls left and rises right.
d. the graph of f(x) falls left and falls right.
b. find the x - intercepts. state whether the graph crosses the x - axis, or touches the x - axis and turns around, at each intercept. what are the x - intercepts?
x = - 3,0
(use a comma to separate answers as needed.)
at which x - intercepts does the graph of the function cross the x - axis? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the graph of the function crosses the x - axis at x = - 3.
(use a comma to separate answers as needed.)
b. there are no x - intercepts at which the graph crosses the x - axis.
at which x - intercepts does the graph of the function touch the x - axis and turn around? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the graph of the function touches the x - axis and turns around at x =
(use a comma to separate answers as needed.)
b. there are no x - intercepts at which the graph touches the x - axis and turns around.

Explanation:

Step1: Analyze end - behavior

For a polynomial function $f(x)=a_nx^n + a_{n - 1}x^{n - 1}+\cdots+a_0$, the end - behavior is determined by the leading term. Here, $f(x)=x^3 + 3x^2$, the leading term is $x^3$ with $n = 3$ (odd) and $a_n=1>0$. As $x\to-\infty$, $y\to-\infty$ (falls left) and as $x\to+\infty$, $y\to+\infty$ (rises right).

Step2: Find x - intercepts

Set $f(x)=0$, so $3x^2 + x^3=x^2(3 + x)=0$. Using the zero - product property, $x^2=0$ gives $x = 0$ and $3 + x=0$ gives $x=-3$.

Step3: Determine crossing/touching at x - intercepts

For a factor of the form $(x - c)^k$ in the factored form of the polynomial:

• If $k$ is odd, the graph crosses the $x$ - axis at $x = c$. Since the factor $(x + 3)$ has $k = 1$ (odd), the graph crosses the $x$ - axis at $x=-3$.
• If $k$ is even, the graph touches the $x$ - axis and turns around at $x = c$. Since the factor $x^2$ has $k = 2$ (even), the graph touches the $x$ - axis and turns around at $x = 0$.

Answer:

a. C. The graph of $f(x)$ falls left and rises right.
b. $x=-3,0$; The graph of the function crosses the $x$ - axis at $x=-3$; The graph of the function touches the $x$ - axis and turns around at $x = 0$.