CCSS.Math.Content.HSF-IF.C.7c
The Standard
Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
Common Core State Standards for Mathematics · Analyze functions using different representations
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students use a polynomial’s factors to find where its graph meets the x-axis. They sketch the graph and show how each tail behaves based on the degree and leading coefficient.
What Mastery Looks Like
- A student can use a factored equation to locate each x-intercept and decide whether the graph crosses or turns there. The sketch shows correct intercepts, a reasonable shape, and correct left and right tail directions.
Common Misconceptions
- Students may report 2 as the zero of x + 2 instead of -2. They often draw every zero as a crossing, even when an even multiplicity makes the graph touch and turn. They may reverse the tails by ignoring the leading coefficient’s sign.
How to Assess It
- Ask students to sketch f(x) = (x + 2)(x - 1)^2, label each zero, and add arrows showing both tails. Have them explain why the graph crosses or turns at each zero.
Lesson moves
Ways to Teach It
Use a floor coordinate grid, place sticky notes at roots, then shape yarn through them and add arrow cards for both ends.
Compare f(x) = (x - 2)^2(x + 1) and g(x) = (x - 2)(x + 1)^2; write how repeated factors change each graph.
Play a card match with factored equations, zero lists, tail descriptions, and graph cards; students justify every completed set.
Graph V(x) = x(20 - 2x)(30 - 2x) for an open box cut from a 20-by-30 sheet, then explain which zeros make physical sense.
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Printable HSF-IF.C.7c Worksheet
A ready-to-print activity worksheet aligned to CCSS.Math.Content.HSF-IF.C.7c, with an answer key for the teacher on its own page. No account needed.
PDF, US Letter, prints cleanly in black and white.
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Where This Standard Sits
Before This Standard
If students are struggling here, check these first.
- CCSS.Math.Content.HSA-APR.B.3
Finding zeros from factored forms and sketching their graph locations are essential parts of graphing polynomials with end behavior.
- CCSS.Math.Content.HSA-SSE.A.2
Recognizing expression structure supports factoring polynomials, which helps identify zeros needed when graphing polynomial functions.
- CCSS.Math.Content.HSA-REI.D.10
Students use the idea that points on y=f(x) satisfy the equation when plotting polynomials and interpreting zeros.
Keep exploring
Related Standards
- CCSS.Math.Content.HSA-APR.B
Understand the relationship between zeros and factors of polynomials
- CCSS.Math.Content.HSF-IF.C.8a
Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in ...
- CCSS.Math.Content.HSF-IF.C.7d
(+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
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