3. which of the following is the graph of (y = (x + 2)^3(x - 3)^2)?

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "End Behavior of Polynomials", "Leading Coefficient Test", "Multiplicity of Roots", "Polynomial Graphing", "Zeros of a Function from Graph" ], "new_concepts": [], "current_concepts": [ "End Behavior of Polynomials", "Leading Coefficient Test", "Multiplicity of Roots", "Polynomial Graphing", "Zeros of a Function from Graph" ] } ## Identify the zeros and their multiplicities Using the *Multiplicity of Roots* knowledge point $$ \begin{aligned} &\text{For } y = (x + 2)^3(x - 3)^2:\ &\text{Root } x = -2 \text{ has multiplicity } 3 \text{ (odd, crosses the x-axis with an inflection point).}\ &\text{Root } x = 3 \text{ has multiplicity } 2 \text{ (even, touches and turns around at the x-axis).} \end{aligned} $$ ## Determine the end behavior of the polynomial Using the *End Behavior of Polynomials* and *Leading Coefficient Test* knowledge points $$ \begin{aligned} &\text{Degree } n = 3 + 2 = 5 \text{ (odd degree).}\ &\text{Leading coefficient } a_n = 1 > 0 \text{ (positive).}\ &\text{As } x \to \infty, y \to \infty.\ &\text{As } x \to -\infty, y \to -\infty. \end{aligned} $$ ## Calculate the y-intercept Using the *Polynomial Graphing* knowledge point $$ \begin{aligned} &y(0) = (0 + 2)^3(0 - 3)^2\ &y(0) = 8 \cdot 9 = 72 \end{aligned} $$ ## Match with the given graph options Using the *Zeros of a Function from Graph* and *Polynomial Graphing* knowledge points $$ \begin{aligned} &\text{The shown graph has:}\ &\text{1. } x\text{-intercepts at } x = -2 \text{ (crosses) and } x = 3 \text{ (touches).}\ &\text{2. } y\text{-intercept at } (0, -72) \text{ instead of } (0, 72). \text{ The graph goes down to } -72 \text{ at } x=0.\ &\text{3. End behavior: } y \to -\infty \text{ as } x \to \infty \text{ and } y \to \infty \text{ as } x \to -\infty.\ &\text{This corresponds to the negative of our function: } y = -(x + 2)^3(x - 3)^2.\ &\text{Therefore, the displayed graph is NOT the graph of } y = (x + 2)^3(x - 3)^2 \text{ (it is its reflection).} \end{aligned} $$

(A) The displayed graph is not the correct graph because its y-intercept is at $(0, -72)$ and its end behavior is reversed; the correct graph must have a y-intercept at $(0, 72)$, rise to infinity as $x \to \infty$, and fall to negative infinity as $x \to -\infty$.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Graphing" ] }

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Question

Explanation:

Identify the zeros and their multiplicities

Determine the end behavior of the polynomial

Calculate the y-intercept

Match with the given graph options

Answer: