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

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Accepted Answer

{ "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", "Multiplicity of Roots", "Polynomial Graphing" ], "new_concepts": [], "current_concepts": [ "Polynomial Graphing", "Multiplicity of Roots", "End Behavior of Polynomials" ] } ## Identify the roots and their multiplicities Using the *Multiplicity of Roots* knowledge point $$ \begin{aligned} &\text{For } y = (x + 2)^3(x - 3)^2:\ &x = -2 \text{ has multiplicity } 3 \quad (\text{odd, crosses the x-axis with an inflection point})\ &x = 3 \text{ has multiplicity } 2 \quad (\text{even, tangent to the x-axis, bounces off}) \end{aligned} $$ ## Determine the end behavior of the function Using the *End Behavior of Polynomials* knowledge point $$ \begin{aligned} &\text{Degree } n = 3 + 2 = 5 \quad (\text{odd})\ &\text{Leading coefficient } a_n = 1 > 0\ &\lim_{x \to -\infty} y = -\infty, \quad \lim_{x \to \infty} y = \infty \end{aligned} $$ ## 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 Using the *Polynomial Graphing* knowledge point $$ \begin{aligned} &\text{The graph crosses at } x = -2 \text{ with a flat inflection point.}\ &\text{The graph bounces off the x-axis at } x = 3.\ &\text{The y-intercept is at } (0, 72).\ &\text{The displayed graph matches all these characteristics.} \end{aligned} $$ ## Question 4 (A) The displayed graph showing an inflection point at $x = -2$, a y-intercept at $(0, 72)$, and a bounce at $x = 3$

(B) A graph with a bounce at $x = -2$ and crossing at $x = 3$

(C) A graph with a y-intercept at $(0, -72)$

(D) A graph with end behavior going from positive infinity to negative infinity

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

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QUESTION IMAGE

Question

Explanation:

Identify the roots and their multiplicities

Determine the end behavior of the function

Calculate the y-intercept

Match with the given graph

Answer:

Question 4