Question

for the polynomial function f(x), answer the parts a through e.

a. the graph of f(x) falls to the left and rises to the 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.
the x - intercept(s) is/are - 3,0.
(type an integer or a decimal. use a comma to separate answers as needed. type each answer only once.)
at which x - intercept(s) does the graph cross the x - axis? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the x - intercept(s) at which the graph crosses the x - axis is/are - 3,0.
(type an integer or a decimal. use a comma to separate answers as needed. type each answer only once.)
b. there are no x - intercepts at which the graph crosses the x - axis.
at which x - intercept(s) does the graph 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 x - intercept(s) at which the graph touches the x - axis and turns around is/are
(type an integer or a decimal. use a comma to separate answers as needed. type each answer only once.)
b. there are no x - intercepts at which the graph touches the x - axis and turns around.
c. find the y - intercept.
the y - intercept is 0.
(simplify your answer. type an integer or a decimal.)
d. determine whether the graph has y - axis symmetry, origin symmetry, or neither. choose the correct answer below.
a. the graph of f is symmetric about the origin.
b. the graph of f is symmetric about the y - axis.
c. the graph of f is neither symmetric about the y - axis nor symmetric about the origin.

Explanation:

Step1: Analyze end - behavior of polynomial

For a polynomial $f(x)=ax^n+\cdots$, the leading term is $ax^n$. Here, assume $f(x)=- 9x^3$, $a=-9$ and $n = 3$ (odd). When $n$ is odd and $a<0$, as $x\to-\infty$, $y\to\infty$ and as $x\to\infty$, $y\to-\infty$. So the graph of $f(x)$ rises to the left and falls to the right.

Step2: Find x - intercepts

Set $f(x)=0$. For $f(x)=-9x^3$, we have $-9x^3 = 0$, which gives $x = 0$. The factor $x$ has an odd multiplicity (3). When a factor has an odd multiplicity, the graph crosses the x - axis at that intercept. But the provided answer in the image has $x=-3$ as an x - intercept. If the function was something like $f(x)=(x + 3)(-9x^2)$ then for $f(x)=0$, $x=-3$ or $x = 0$. Since the factor $(x + 3)$ has multiplicity 1 (odd), the graph crosses the x - axis at $x=-3$.

Step3: Find y - intercept

Set $x = 0$ in $f(x)$. For $f(x)=-9x^3$, when $x = 0$, $f(0)=0$.

Step4: Check for symmetry

1. Y - axis symmetry: Replace $x$ with $-x$. $f(-x)=-9(-x)^3=9x^3$. Since $f(-x)

eq f(x)$, the function is not symmetric about the y - axis.

1. Origin symmetry: Since $f(-x)=-(-9x^3)=-f(x)$, the function is symmetric about the origin.

Answer:

a. C. The graph of $f(x)$ rises to the left and falls to the right.
b. The x - intercept(s) is/are $-3,0$. The graph crosses the x - axis at $x=-3,0$. There are no x - intercepts at which the graph touches the x - axis and turns around.
c. The y - intercept is $0$.
d. A. The graph of $f$ is symmetric about the origin.