CCSS.Math.Content.HSA-REI.D.11
The Standard
Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
Common Core State Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students connect a graph intersection to an input that gives both functions the same output. They estimate those inputs from graphs, tables, or repeated numerical trials across several function types.
What Mastery Looks Like
- Given an interval, a student locates every intersection and reports the x-values to a stated precision. The student explains that both rules give the same y-value there and checks each estimate by substitution.
Common Misconceptions
- Students may report the full ordered pair when only the input is requested. They may miss tangencies, assume there is one solution, or mistake an asymptote for a crossing. A poor graphing window can also hide solutions.
How to Assess It
- Exit ticket: Graph y = x + 2 and y = x² − 2. Estimate each intersection’s x-value to the nearest tenth and explain why it solves x + 2 = x² − 2.
Lesson moves
Ways to Teach It
Give pairs of function tables; students plot both on one grid, circle each crossing, and record the matching input.
Prompt: Explain why a shared point makes f(x) = g(x), and why the solution is the point’s x-value.
Run a card match with function pairs, graph screenshots, and approximate solution sets; students justify each match before keeping it.
Compare two phone plans, C = 25 + 10m and C = 55 + 5m; graph them and find when their costs are equal.
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Printable HSA-REI.D.11 Worksheet
A ready-to-print activity worksheet aligned to CCSS.Math.Content.HSA-REI.D.11, 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-REI.D.10
Students must know each graph shows all solutions to its equation to see intersections as inputs making both outputs equal.
- CCSS.Math.Content.8.F.A.1
Students must know graphs pair each input with output to see intersections as equal outputs for the same x-value.
- CCSS.Math.Content.8.EE.C.8a
The idea that an intersection satisfies both equations extends from linear systems to solving f(x)=g(x) graphically.
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