consider the following polynomial function.

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

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:

Question

consider the following polynomial function.

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

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:

Accepted Answer

{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Polynomial Degree", "Leading Coefficient Sign", "Parabola Intercepts" ], "new_concepts": [ "Polynomial End Behavior", "Multiplicity of Zeros" ], "current_concepts": [ "Polynomial Degree", "Leading Coefficient Sign", "Polynomial End Behavior", "Multiplicity of Zeros", "Parabola Intercepts" ] } ## Determine the end behavior of the graph of $f$ $$ \begin{aligned} &f(x) = (x+1)^2(x-2) = (x^2 + 2x + 1)(x-2) = x^3 - 3x - 2\ &\text{Degree } n = 3 \quad (\text{odd})\ &\text{Leading coefficient } a_n = 1 > 0\ &\lim_{x \to -\infty} f(x) = -\infty \quad \text{and} \quad \lim_{x \to \infty} f(x) = \infty \end{aligned} $$ ## Find the real zeros and analyze their multiplicity $$ \begin{aligned} &f(x) = 0 \implies (x+1)^2(x-2) = 0\ &x = -1 \quad (\text{multiplicity } 2 \implies \text{touches but does not cross})\ &x = 2 \quad (\text{multiplicity } 1 \implies \text{crosses}) \end{aligned} $$ ## Calculate the $y$-intercept $$ \begin{aligned} &y = f(0) = (0+1)^2(0-2) = 1 \cdot (-2) = -2 \end{aligned} $$ ## Question 1 (a) Choose the end behavior of the graph of $f$: Falls to the left and rises to the right. ## Question 2 (b) List each real zero of $f$ according to the behavior of the graph at the $x$-axis near that zero: - Zero(s) where the graph crosses the $x$-axis: 2 - Zero(s) where the graph touches, but does not cross the $x$-axis: -1 ## Question 3 (c) Find the $y$-intercept of the graph of $f$: $y$-intercept: -2 { "elements": [ { "type": "functiongraph", "params": [ { "js": "Math.pow(x+1, 2)*(x-2)", "latex": "(x+1)^2(x-2)" }, -3, 4 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "f(x)", "withLabel": true } }, { "type": "point", "params": [ [-1, 0] ], "properties": { "name": "(-1, 0)", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "point", "params": [ [2, 0] ], "properties": { "name": "(2, 0)", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "point", "params": [ [0, -2] ], "properties": { "name": "(0, -2)", "size": 4, "color": "#5583F2", "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 of \(f\)

Find the real zeros and analyze their multiplicity

Calculate the \(y\)-intercept

Answer:

Question 1

Question 2

Question 3