complete the statements about the key features of the graph of $f(x) = x^5 - 9x^3$.

as $x$ goes to negative infinity, $f(x)$ goes to negative infinity, and as $x$ goes to positive infinity, $f(x)$ goes to positive infinity.

choose all of the zeroes of $f(x)$.
-3 with multiplicity 1
3 with multiplicity 1
0 with multiplicity 1
0 with multiplicity 3
3 with multiplicity 0

the graph of $f(x)$ touches but does not cross / crosses / neither crosses nor touches the $x$-axis at $x = -3, 0,$ and 3.

Question

complete the statements about the key features of the graph of $f(x) = x^5 - 9x^3$.

as $x$ goes to negative infinity, $f(x)$ goes to negative infinity, and as $x$ goes to positive infinity, $f(x)$ goes to positive infinity.

choose all of the zeroes of $f(x)$.
-3 with multiplicity 1
3 with multiplicity 1
0 with multiplicity 1
0 with multiplicity 3
3 with multiplicity 0

the graph of $f(x)$ touches but does not cross / crosses / neither crosses nor touches the $x$-axis at $x = -3, 0,$ and 3.

Accepted Answer

{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [], "current_concepts": [ "End Behavior", "Root Multiplicity", "Polynomial Graph Behavior" ] } ## Determine end behavior of $f(x) = x^5 - 9x^3$ $$ \begin{aligned} &\text{Degree: } n = 5 \quad (\text{odd})\ &\text{Leading coefficient: } a_n = 1 > 0\ &\lim_{x \to -\infty} f(x) = -\infty\ &\lim_{x \to \infty} f(x) = \infty \end{aligned} $$ ## Find and analyze zeroes of $f(x)$ $$ \begin{aligned} &f(x) = x^3(x^2 - 9) = x^3(x - 3)(x + 3)\ &x = 0 \quad (\text{multiplicity } 3)\ &x = 3 \quad (\text{multiplicity } 1)\ &x = -3 \quad (\text{multiplicity } 1) \end{aligned} $$ ## Determine graph behavior at the x-axis $$ \begin{aligned} &\text{Multiplicities at } x = -3, 0, 3 \text{ are all odd } (1, 3, 1)\ &\text{Since all multiplicities are odd, the graph crosses the x-axis at all three zeroes} \end{aligned} $$ ## Question 1 As $x$ goes to negative infinity, $f(x)$ goes to negative infinity, and as $x$ goes to positive infinity, $f(x)$ goes to positive infinity. ## Question 2 Choose all of the zeroes of $f(x)$. - [x] $-3$ with multiplicity 1 - [x] $3$ with multiplicity 1 - [ ] $0$ with multiplicity 1 - [x] $0$ with multiplicity 3 - [ ] $3$ with multiplicity 0 ## Question 3 The graph of $f(x)$ crosses the x-axis at $x = -3, 0, \text{ and } 3$. { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Graph Behavior" ] }

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Question

Explanation:

Determine end behavior of \(f(x) = x^5 - 9x^3\)

Find and analyze zeroes of \(f(x)\)

Determine graph behavior at the x-axis

Answer:

Question 1

Question 2

Question 3