Question

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

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.

e. if necessary, find a few additional points and graph the function. use the maximum number of turning points to check whether it is drawn correctly. choose the correct graph below.

Explanation:

Determine the symmetry of the function

Using the Symmetry of Functions and Even and Odd Functions knowledge points

$$
LATEXBLOCK0
$$

Analyze the end behavior of the polynomial

Using the End Behavior of Polynomials knowledge point

$$
LATEXBLOCK1
$$

The leading term is \(-x^4\). Since the degree is even (\(4\)) and the leading coefficient is negative (\(-1\)), the graph falls to both the left and the right:

$$ \lim_{x \to \pm\infty} f(x) = -\infty $$

Analyze the behavior at the zeros

Using the Multiplicity of Zeros knowledge point
The zeros of \(f(x) = -x^2(x-3)(x+5)\) are:

• \(x = 0\) with multiplicity \(2\) (the graph is tangent to the \(x\)-axis, touching and turning around at \(x = 0\)).
• \(x = 3\) with multiplicity \(1\) (the graph crosses the \(x\)-axis).
• \(x = -5\) with multiplicity \(1\) (the graph crosses the \(x\)-axis).

Match the correct graph

Using the End Behavior of Polynomials and Multiplicity of Zeros knowledge points

• The graph must cross the \(x\)-axis at \(x = -5\) and \(x = 3\).
• The graph must touch and turn around at the origin \((0,0)\).
• Both ends of the graph must point downwards.
• Evaluating a test point: for \(x = -2\), \(f(-2) = -(-2)^2(-2-3)(-2+5) = -4(-5)(3) = 60 > 0\).
• Evaluating another test point: for \(x = 2\), \(f(2) = -(2)^2(2-3)(2+5) = -4(-1)(7) = 28 > 0\).
• This matches the graph in Option B, which has local maxima on both sides of the origin and touches the origin from above.

Answer:

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)

Question e

• A. Graph with a local minimum at the origin and falling ends.
• B. Graph with a local minimum touching the origin from above, crossing at -5 and 3, and falling ends. (Correct answer)
• C. Graph with rising ends.
• D. Graph with a local maximum at the origin and falling ends.