use the leading coefficient test to determine the end behavior of the polynomial function. then use this end behavior to match the function with its graph.

$f(x) = -6x^3 + 2x^2 + 3x - 2$

a. falls to the left and rises to the right
b. falls to the left and falls to the right
c. rises to the left and rises to the right
d. rises to the left and falls to the right

Question

use the leading coefficient test to determine the end behavior of the polynomial function. then use this end behavior to match the function with its graph.

$f(x) = -6x^3 + 2x^2 + 3x - 2$

a. falls to the left and rises to the right
b. falls to the left and falls to the right
c. rises to the left and rises to the right
d. rises to the left and falls to the right

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": [ "Leading Coefficient Test", "End Behavior of Polynomials" ], "new_concepts": [], "current_concepts": [ "Leading Coefficient Test", "End Behavior of Polynomials", "Degree of Polynomial" ] } ## Identify the leading term Using the *Leading Coefficient Test* and *End Behavior of Polynomials* knowledge points $$ \begin{aligned} &f(x) = -6x^3 + 2x^2 + 3x - 2\ &\text{Leading term: } -6x^3 \end{aligned} $$ ## Determine degree and leading coefficient Using the *Degree of Polynomial* and *Leading Coefficient Test* knowledge points $$ \begin{aligned} &n = 3 \quad (\text{odd degree})\ &a_n = -6 \quad (\text{negative leading coefficient}) \end{aligned} $$ ## Determine end behavior Using the *End Behavior of Polynomials* knowledge point $$ \begin{aligned} &\lim_{x \to -\infty} f(x) = \infty \quad (\text{rises to the left})\ &\lim_{x \to \infty} f(x) = -\infty \quad (\text{falls to the right}) \end{aligned} $$ ## Match with the correct graph Using the *End Behavior of Polynomials* knowledge point $$ \begin{aligned} &\text{Graph D rises on the left and falls on the right.}\ &\text{Y-intercept: } f(0) = -2 \quad (\text{matches Graph D}) \end{aligned} $$

(A) falls to the left and rises to the right

(B) falls to the left and falls to the right

(C) rises to the left and rises to the right

(D) rises to the left and falls to the right

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "End Behavior of Polynomials" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the leading term

Determine degree and leading coefficient

Determine end behavior

Match with the correct graph

Answer: