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

a. use the leading coefficient test to determine the graphs end behavior

a. the graph of $f(x)$ rises to the left and falls to the right.
b. the graph of $f(x)$ falls to the left and rises to the right.
c. the graph of $f(x)$ falls to the left and falls to the right.
d. the graph of $f(x)$ rises to the left and rises to the right.

b. find the x-intercept(s).

the x-intercept(s) is/are $0, 3, -5$
(type an integer or a decimal. use a comma to separate answers as needed.)

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)
(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.

Question

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

a. use the leading coefficient test to determine the graphs end behavior

a. the graph of $f(x)$ rises to the left and falls to the right.
b. the graph of $f(x)$ falls to the left and rises to the right.
c. the graph of $f(x)$ falls to the left and falls to the right.
d. the graph of $f(x)$ rises to the left and rises to the right.

b. find the x-intercept(s).

the x-intercept(s) is/are $0, 3, -5$
(type an integer or a decimal. use a comma to separate answers as needed.)

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)
(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.

Accepted Answer

{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Leading Coefficient Test", "End Behavior of Polynomials", "Multiplicity of Zeros", "Graph Behavior at Zeros" ], "new_concepts": [], "current_concepts": [ "Leading Coefficient Test", "End Behavior of Polynomials", "Multiplicity of Zeros", "Graph Behavior at Zeros" ] } ## Determine the degree and leading coefficient Using the *Leading Coefficient Test* and *End Behavior of Polynomials* knowledge points $$ \begin{aligned} &f(x) = -x^2(x-3)(x+5)\ &\text{Degree } n = 2 + 1 + 1 = 4 \quad (\text{even})\ &\text{Leading coefficient } a_n = -1 \quad (\text{negative}) \end{aligned} $$ ## Determine the end behavior Using the *Leading Coefficient Test* and *End Behavior of Polynomials* knowledge points $$ \begin{aligned} &\lim_{x \to -\infty} f(x) = -\infty\ &\lim_{x \to \infty} f(x) = -\infty \end{aligned} $$ The graph falls to the left and falls to the right. ## Find the x-intercepts Using the *Multiplicity of Zeros* knowledge point $$ \begin{aligned} &-x^2(x-3)(x+5) = 0\ &x = 0 \quad (\text{multiplicity } 2)\ &x = 3 \quad (\text{multiplicity } 1)\ &x = -5 \quad (\text{multiplicity } 1) \end{aligned} $$ ## Determine where the graph crosses the x-axis Using the *Graph Behavior at Zeros* and *Multiplicity of Zeros* knowledge points $$ \begin{aligned} &\text{At } x = 0: \text{ multiplicity is even (2)} \implies \text{touches and turns}\ &\text{At } x = 3: \text{ multiplicity is odd (1)} \implies \text{crosses}\ &\text{At } x = -5: \text{ multiplicity is odd (1)} \implies \text{crosses} \end{aligned} $$ The graph crosses the x-axis at the zeros $3, -5$. ## Question a A. The graph of f(x) rises to the left and falls to the right.

B. The graph of f(x) falls to the left and rises to the right.

C. The graph of f(x) falls to the left and falls to the right.

D. The graph of f(x) rises to the left and rises to the right. ## Question b The x-intercept(s) is/are 0, 3, -5 ## Question c A. The graph crosses the x-axis at the zero(s) 3, -5

B. There are no zeros at which the graph crosses the x-axis

{ "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "End Behavior of Polynomials" ] }

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QUESTION IMAGE

Question

Explanation:

Determine the degree and leading coefficient

Determine the end behavior

Find the x-intercepts

Determine where the graph crosses the x-axis

Answer:

Question a

Question b

Question c