Montana Math MT.PLUS.ALG.FUN.1
The Standard
Identify the effect on the graph of replacing f(x)f(x) by f(x) + k, k · f(x), f(k · x), and f(x + k) for specific values of k (both positive and negative).
2026 Montana Mathematics Content Standards · Algebraic and Functional Reasoning
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students use a known graph to predict how changes to its formula move or reshape it. They sketch the new graph and track points through shifts, stretches, compressions, and reflections.
What Mastery Looks Like
- Given a graph and a value of k, students sketch the changed graph and show where a point moves. They can explain why f(x+2) moves left two units and f(-x) reflects across the y-axis.
Common Misconceptions
- Students often move f(x+2) right instead of left. They may treat 2f(x) and f(2x) as the same change. They may reflect across the wrong axis when the multiplier is negative.
How to Assess It
- Give students f(x)=x²+x and the point (1,2). Ask where that point moves on f(x)+2, -2f(x), f(-2x), and f(x-2), and have them name each change.
Lesson moves
Ways to Teach It
Use tracing paper to copy f(x)=x²+x, then slide it two squares left and flip it across each axis.
Ask pairs, 'Why does f(x+2) shift left while f(x)+2 shifts up?' Have them use the point (1,2) to explain.
Deal cards showing f(x)-3, 2f(x), f(2x), and f(x-3); students race to match each card with a printed sketch.
Use h(t)=-t²+4t for a ball's height, then sketch h(t)+2 to model a launch from a two-metre platform.
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Related Standards
- MT.PLUS.ALG.EXP.3.a
Expressing exponential functions in the form f(x) = a · b^x and f(x) = Pe^(r·t), and
- MT.PLUS.ALG.EXP.4
Graph logarithmic and exponential functions with and without the use of technology by identifying intercepts, asymptotes, and end behavior.
- MT.8.F.2
Compare properties of two functions using tables, graphs, and equations.
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