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
What this standard means
Students need to graph polynomial functions from factored or factorable forms. They should find zeros, plot intercepts, use multiplicity to decide whether the graph crosses or bounces, and describe what happens on the far left and far right.
Mastery looks like a clear sketch with labeled zeros, correct crossing or touching behavior, a y-intercept when useful, and correct end behavior from degree and leading coefficient. Students often mix up multiplicity, forget repeated roots, or think every polynomial alternates perfectly between zeros. They also may rely on calculators without explaining why the graph has that shape.
Ways to teach it
- Use algebra tiles or graphing paper to factor simple polynomials, mark zeros on a number line, then sketch the graph behavior around each zero.
- Ask students: How does changing one factor from x minus 2 to squared affect the graph at x equals 2?
- Give three factored polynomials and have students identify zeros, multiplicities, end behavior, and a rough sketch in five minutes.
- Connect to projectile or profit models by asking where the graph crosses the x-axis and what those zeros mean in context.
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Related standards
- CCSS.Math.Content.HSA-APR.B
Understand the relationship between zeros and factors of polynomials
- CCSS.Math.Content.HSA-APR.B.3
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial...
- 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.