Question

for the polynomial function \\(f(x) = -x^2(x - 3)(x + 5)\\), answer parts a through e.

at which zero(s) 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 crosses the x-axis at the zero(s) 3, -5.
(type an integer or a decimal. use a comma to separate answers as needed.)
b. there are no zeros at which the graph crosses the x-axis.

at which zero(s) 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 touches the x-axis and turns around at the zero(s) 0.
(type an integer or a decimal. use a comma to separate answers as needed.)
b. there are no zeros at which the graph touches the x-axis and turns around.

c. find the y-intercept.

the y-intercept is 0.
(simplify your answer.)

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 y-axis.
b. the graph of f is symmetric about the origin.
c. the graph of f is neither symmetric about the y-axis nor symmetric about the origin.

Explanation:

Find zeros and multiplicities

Using the Multiplicity of Zeros knowledge point

$$
LATEXBLOCK0
$$

Determine x-axis crossing behavior

Using the Graph Behavior at Zeros knowledge point

$$
LATEXBLOCK1
$$

Find the y-intercept

Evaluate the function at \(x = 0\):

$$ f(0) = -(0)^2(0-3)(0+5) = 0 $$

Thus, the \(y\)-intercept is \(0\).

Determine function symmetry

We test for symmetry by evaluating \(f(-x)\):

$$
LATEXBLOCK2
$$

Since \(f(-x)
eq f(x)\) and \(f(-x)
eq -f(x)\), the graph is neither symmetric about the \(y\)-axis nor symmetric about the origin.

Answer:

Question a

• A. The graph crosses the x-axis at the zero(s) 3, -5 (Correct answer)
• B. There are no zeros at which the graph crosses the x-axis.

Question b

• A. The graph touches the x-axis and turns around at the zero(s) 0 (Correct answer)
• B. There are no zeros at which the graph touches the x-axis and turns around.

Question c

The y-intercept is <blank>0</blank>.

Question d

• A. The graph of f is symmetric about the y-axis.
• B. The graph of f is symmetric about the origin.
• C. The graph of f is neither symmetric about the y-axis nor symmetric about the origin. (Correct answer)