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
What this standard means
Students need to connect changes in a function rule to changes in its graph. They should describe vertical and horizontal shifts, stretches, compressions, and reflections using specific values of k. They also need to work backward from two graphs to find the value of k. Technology should be used to test and explain patterns.
Mastery looks like a student saying, “f(x) + 3 moves the graph up 3,” or “f(2x) makes it narrower,” and justifying it with points or a graph. Common trouble spots are mixing up f(x + k) with f(x) + k, misunderstanding negative k values, and not seeing why even graphs have y-axis symmetry while odd graphs have origin symmetry.
Ways to teach it
- Hands-on: Use Desmos sliders for y=f(x)+k, y=kf(x), y=f(kx), and y=f(x+k), then record one pattern per slider.
- Prompt: Given y=x^2 and y=(x-4)^2+2, write how the graph moved and name the changes in order.
- Quick assessment: Show two graphs of the same function, and ask students to find k for f(x)+k or f(x+k).
- Real-world connection: Model a phone plan cost graph, then ask how adding a setup fee or doubling the rate changes the graph.
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