consider the following polynomial function.

\f(x) = -(x - 1)(x + 2)(x + 1)^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)(x + 2)(x + 1)^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 End Behavior", "Multiplicity of Zeros" ], "new_concepts": [ "Polynomial Graphing" ], "current_concepts": [ "Polynomial End Behavior", "Multiplicity of Zeros", "Polynomial Graphing" ] } ## Determine degree and leading coefficient $$ \begin{aligned} &f(x) = -(x - 1)^1(x + 2)^1(x + 1)^2 \ &\text{Degree } n = 1 + 1 + 2 = 4 \quad (\text{even}) \ &\text{Leading coefficient } a_n = -1 \quad (\text{negative}) \ &\lim_{x \to \infty} f(x) = -\infty, \quad \lim_{x \to -\infty} f(x) = -\infty \end{aligned} $$ ## Find zeros and analyze multiplicity $$ \begin{aligned} &x - 1 = 0 \implies x = 1 \quad (\text{multiplicity } 1, \text{ odd } \implies \text{crosses}) \ &x + 2 = 0 \implies x = -2 \quad (\text{multiplicity } 1, \text{ odd } \implies \text{crosses}) \ &x + 1 = 0 \implies x = -1 \quad (\text{multiplicity } 2, \text{ even } \implies \text{touches}) \end{aligned} $$ ## Calculate y-intercept $$ \begin{aligned} f(0) &= -(0 - 1)(0 + 2)(0 + 1)^2 \ &= -(-1)(2)(1)^2 \ &= 2 \end{aligned} $$ ## Question (a) The graph of $f$ falls to the left and falls to the right. ## Question (b) - Zero(s) where the graph crosses the $x$-axis: $-2, 1$ - Zero(s) where the graph touches, but does not cross the $x$-axis: $-1$ ## Question (c) - $y$-intercept: $2$ { "elements": [ { "type": "functiongraph", "params": [ { "js": "-(x-1)*(x+2)*Math.pow(x+1, 2)", "latex": "-(x-1)(x+2)(x+1)^2" }, -3, 2 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3 } }, { "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": [[-1, 0]], "properties": { "name": "(-1,0)", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "point", "params": [[0, 2]], "properties": { "name": "(0,2)", "size": 4, "color": "#5583F2", "withLabel": true } } ] } { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Graphing" ] }

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QUESTION IMAGE

Question

Explanation:

Determine degree and leading coefficient

Find zeros and analyze multiplicity

Calculate y-intercept

Answer:

Question (a)

Question (b)

Question (c)