Polynomial Functions and End Behavior

Develop degree and leading term, even and odd degree, interpreting graphs.

Lesson 1 of 43 parts Practice Flashcards

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Degree and Leading Term

Objective: Explain degree and leading term, apply the rule, and justify relevant restrictions.

The highest nonzero power and its coefficient dominate polynomial behavior for large |x|. The degree does not depend on smaller terms. Their relative influence shrinks as |x| grows. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$p(x)=aₙxⁿ+…+a₀; aₙ≠0$

Worked examples

Try itWhat is the degree of 3x^7+5x²+1 (c>2)?Show answer

Step 1: Identify the greatest nonzero exponent. Step 2: The x^7 term has coefficient 3≠0. Step 3: The degree is 7.

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Degree and Leading Term

Objective: Explain degree and leading term, apply the rule, and justify relevant restrictions.

Formulas

$p(x)=aₙxⁿ+…+a₀; aₙ≠0$

Worked examples

Try itWhat is the degree of 3x^7+5x²+1 (c>2)?Show answer

Step 1: Identify the greatest nonzero exponent. Step 2: The x^7 term has coefficient 3≠0. Step 3: The degree is 7.

Even and Odd Degree

Objective: Explain even and odd degree, apply the rule, and justify relevant restrictions.

A polynomial of even degree has ends heading the same way; an odd-degree polynomial has ends heading opposite ways, with direction set by its leading sign. No, the leading term controls them. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

Leading term aₙxⁿ determines polynomial end behavior

Worked examples

Try itFor p(x)=−3x^15+5, is its right-end output positive or negative? Answer 1 for positive, −1 for negative.Show answer

Step 1: The degree 15 is odd. Step 2: The leading coefficient −3 is negative. Step 3: As x→+∞, the output tends to −∞; choose −1.

Interpreting Graphs

Objective: Explain interpreting graphs, apply the rule, and justify relevant restrictions.

A polynomial graph is continuous and smooth in the usual classroom model; degree bounds the number of real zeros and turning points. No, it has at most four distinct zeros. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$p(x)=aₙxⁿ+…+a₀; aₙ≠0$

Worked examples

Try itWhat is the degree of 3x^7+5x²+1 (c>2)?Show answer

Step 1: Identify the greatest nonzero exponent. Step 2: The x^7 term has coefficient 3≠0. Step 3: The degree is 7.

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