\f(x) = -x^2(x-2)^2(x+1)\

answer the questions regarding the graph of $f$.
then, use this information to graph the function.

(a) choose the end behavior of the graph of $f$.

(b) list each real zero of $f$ according to the behavior of the graph at the $x$-axis near that zero. if there is more than one answer, separate them with commas. if there is no answer, click on
one\.

zero(s) where the graph crosses the $x$-axis:
zero(s) where the graph touches, but does not cross the $x$-axis:

(c) find the $y$-intercept of the graph of $f$.

$y$-intercept:

(d) graph $f(x) = -x^2(x-2)^2(x+1)$ by doing the following.

Question

\f(x) = -x^2(x-2)^2(x+1)\

answer the questions regarding the graph of $f$.
then, use this information to graph the function.

(a) choose the end behavior of the graph of $f$.

(b) list each real zero of $f$ according to the behavior of the graph at the $x$-axis near that zero. if there is more than one answer, separate them with commas. if there is no answer, click on
one\.

zero(s) where the graph crosses the $x$-axis:
zero(s) where the graph touches, but does not cross the $x$-axis:

(c) find the $y$-intercept of the graph of $f$.

$y$-intercept:

(d) graph $f(x) = -x^2(x-2)^2(x+1)$ by doing the following.

Accepted Answer

{ "quality": "clear", "question_count": 4, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Polynomial End Behavior", "Multiplicity of Zeros" ], "new_concepts": [], "current_concepts": [ "Polynomial End Behavior", "Multiplicity of Zeros", "Polynomial Intercepts", "Graphing Polynomials" ] } ## Determine the end behavior of the graph $$ \begin{aligned} f(x) &= -x^2(x-2)^2(x+1) \ \text{Leading term} &= -x^2 \cdot x^2 \cdot x = -x^5 \ \text{Degree } n &= 5 \quad (\text{odd}) \ \text{Leading coefficient } a_n &= -1 \quad (\text{negative}) \ \lim_{x \to \infty} f(x) &= -\infty \ \lim_{x \to -\infty} f(x) &= \infty \end{aligned} $$ ## Find the zeros and analyze their multiplicities $$ \begin{aligned} f(x) &= 0 \implies -x^2(x-2)^2(x+1) = 0 \ x &= 0 \quad (\text{multiplicity } 2, \text{ even}) \implies \text{touches but does not cross} \ x &= 2 \quad (\text{multiplicity } 2, \text{ even}) \implies \text{touches but does not cross} \ x &= -1 \quad (\text{multiplicity } 1, \text{ odd}) \implies \text{crosses} \end{aligned} $$ ## Find the y-intercept $$ \begin{aligned} y &= f(0) = -(0)^2(0-2)^2(0+1) = 0 \end{aligned} $$ ## Question (a) Choose the end behavior of the graph of $f$: Falls to the right and rises to the left (i.e., as $x \to \infty$, $f(x) \to -\infty$, and as $x \to -\infty$, $f(x) \to \infty$). ## Question (b) - Zero(s) where the graph crosses the $x$-axis: -1 - Zero(s) where the graph touches, but does not cross the $x$-axis: 0, 2 ## Question (c) - $y$-intercept: 0 { "elements": [ { "type": "functiongraph", "params": [ { "js": "-x*x * Math.pow(x-2, 2) * (x+1)", "latex": "-x^2(x-2)^2(x+1)" }, -2.0, 3.0 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3 } }, { "type": "point", "params": [[-1, 0]], "properties": { "name": "(-1, 0)", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "point", "params": [[0, 0]], "properties": { "name": "(0, 0)", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "point", "params": [[2, 0]], "properties": { "name": "(2, 0)", "size": 4, "color": "#F2557F", "withLabel": true } } ], "timestamps": [0.5, 1.0, 1.5] } { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial End Behavior" ] }

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QUESTION IMAGE

Question

Explanation:

Determine the end behavior of the graph

Find the zeros and analyze their multiplicities

Find the y-intercept

Answer:

Question (a)

Question (b)

Question (c)