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

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) 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)
(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.

Question

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

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) 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)
(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.

Accepted Answer

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

B. There are no zeros at which the graph touches the x-axis and turns around.

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

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QUESTION IMAGE

Question

Explanation:

Determine the end behavior

Find the x-intercepts

Analyze behavior at zeros (crossing)

Analyze behavior at zeros (touching)

Answer:

Question a

Question b

Question c

Question d