CCSS.Math.Content.HSA-APR.B.3
The Standard
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.
Common Core State Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students set each factor equal to zero and solve for the input values where the polynomial equals zero. They use those values, repeated factors, and end behavior to sketch the graph's overall shape.
What Mastery Looks Like
- Students correctly find every zero from a factored polynomial and note repeated zeros. Their sketch places the x-intercepts correctly and shows whether the graph crosses or touches at each one. The ends point in directions that match the degree and leading coefficient.
Common Misconceptions
- Students may name a factor, such as (x - 3), instead of giving the zero x = 3. They may confuse zeros with the y-intercept. They may also draw the graph crossing at every zero, even when a zero has even multiplicity.
How to Assess It
- Exit ticket: For f(x) = (x + 2)(x - 1)^2, list the zeros and sketch a rough graph. Label where the graph crosses or touches the x-axis.
Lesson moves
Ways to Teach It
Mark zeros on a floor number line, then use yarn to model a graph crossing or touching at each marked value.
Compare graphs of (x - 2)^2(x + 1) and (x - 2)(x + 1)^2, then explain how repeated factors change each graph.
Play a matching game with cards showing factored polynomials, zero lists, multiplicities, and rough graph sketches.
Model profit with P(x) = (x - 2)(x - 8), then identify break-even values and sketch where profit is positive or negative.
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Printable HSA-APR.B.3 Worksheet
A ready-to-print activity worksheet aligned to CCSS.Math.Content.HSA-APR.B.3, 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.2
The Remainder Theorem connects factors to zeros, which supports finding intercepts from factored polynomials for rough graphs.
- CCSS.Math.Content.HSA-SSE.A.2
Seeing polynomial structure supports factoring and reading factors as zeros, which then informs the rough graph.
- CCSS.Math.Content.HSF-IF.B.4
Interpreting intercepts and other graph features supports using polynomial zeros as x-intercepts in a rough graph.
What This Unlocks
Mastery here sets students up for these next.
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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