CCSS.Math.Content.HSF-BF.B.3
The Standard
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
Common Core State Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students describe vertical and horizontal shifts, stretches, compressions, and reflections using graphs and equations. They determine the value and sign of k from a transformed graph. They also identify even, odd, and neither functions using symmetry and algebra.
What Mastery Looks Like
- Given f and a transformed graph, a student writes the matching expression and finds k, including negative values. Their sketches move key points correctly and distinguish vertical scale factors from reciprocal horizontal scale factors. They verify evenness with f(-x)=f(x) and oddness with f(-x)=-f(x).
Common Misconceptions
- Students often think f(x+k) shifts right because the sign is positive. They may treat f(kx) as a horizontal stretch by k instead of a scale by 1/|k|. They may miss reflections from negative values or confuse even symmetry with odd symmetry.
How to Assess It
- Exit ticket: For f(x)=x^3-x, have students sketch g(x)=-f(x+2), describe both transformations, and classify f as even, odd, or neither.
Lesson moves
Ways to Teach It
Trace a parent graph on patty paper, then slide, reflect, and stretch it to match four transformation cards.
Ask students to explain why f(x+3) moves left while f(x)+3 moves up, using one point as evidence.
Run a Desmos slider challenge where students adjust k to match a target graph, then record the transformation.
Graph hourly temperature data, then model a sensor offset with f(x)+k and a delayed reading with f(x-k).
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Printable HSF-BF.B.3 Worksheet
A ready-to-print activity worksheet aligned to CCSS.Math.Content.HSF-BF.B.3, with an answer key for the teacher on its own page. No account needed.
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Where This Standard Sits
Before This Standard
If students are struggling here, check these first.
- CCSS.Math.Content.HSF-IF.A.2
Students must understand f(x) as output for an input before interpreting f(x)+k, f(kx), and f(x+k) transformations.
- CCSS.Math.Content.HSF-IF.A.1
Transforming f(x) requires understanding function notation, inputs, outputs, and the graph y = f(x).
- CCSS.Math.Content.HSF-IF.C.7
Graphing symbolic functions and reading key features supports seeing how shifts, stretches, and reflections change a function’s graph.
What This Unlocks
Mastery here sets students up for these next.
Keep exploring
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