Maryland Math F.IF.C.7.g

MathGrades 9–12INTERPRETING FUNCTIONS

The Standard

F.IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Determine the end behavior of the graph of a polynomial function using the degree and leading coefficient.

Maryland College and Career Ready Standards for Mathematics (2025) · Analyze functions using different representations.

Teacher's field guide

What Students Need to Do: Students find the term with the greatest exponent and check whether its coefficient is positive or negative. They use that term to predict where the graph points far left and far right.
What Mastery Looks Like: For -3x^4+2x, they say both ends fall. For 2x^3-5, they say the left end falls and the right end rises, then sketch both graphs' tails.
Common Misconceptions: Students may use the constant term or the first term written instead of the highest-power term. Some think a negative leading coefficient always makes both ends fall, even when the degree is odd.
How to Assess It: Exit ticket: For f(x)=-2x^3+5x^2-1, describe what happens to the graph far left and far right. Name the term you used to decide.

Lesson moves

Have pairs bend pipe cleaners into the four possible tail patterns, then label each with a sample leading term.
Ask students to explain in two sentences why 4x^5-100x^2 and 4x^5+100x^2 have the same end behavior.
Put eight polynomial cards on desks; partners sort them into four tail-pattern piles, then check each leading term.
Show a profit model, P(x)=-x^4+8x^2, where x is units sold; ask which tail applies to possible sales.

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