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Transformations of Functions for the PSAT/NMSQT: Complete Study Guide + Free Practice Problems
Everything the PSAT/NMSQT tests about function transformations in one place: vertical shifts, horizontal shifts, stretches and compressions, reflections, and combining multiple transformations — with step-by-step examples, worked problems, and a free practice quiz.
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Free · No signupTransformations of Functions
Shifts, stretches, compressions, and reflections.
What is a function transformation?
A transformation takes the graph of a known "parent" function and shifts, stretches, compresses, or reflects it, without changing its basic shape. Every transformation shows up as a constant added, subtracted, or multiplied somewhere in the function's equation.
The PSAT usually gives you the parent function's graph or equation and asks you to identify or predict the result of a transformation — you rarely need to graph anything from scratch.
Vertical shifts
If g(x) = f(x) − 4, how does the graph of g compare to the graph of f?
Horizontal shifts
If g(x) = f(x + 5), how does the graph of g compare to the graph of f?
Horizontal shifts move opposite the sign inside the parentheses. f(x + 5) shifts the graph left, even though the sign inside is positive — this is the single most commonly reversed rule in this topic.
Stretches & compressions
| Form | Effect |
|---|---|
| a · f(x), |a| > 1 | Vertical stretch — graph becomes taller/steeper |
| a · f(x), 0 < |a| < 1 | Vertical compression — graph becomes flatter/shorter |
If g(x) = 3f(x), how does the graph of g compare to the graph of f?
Reflections
| Form | Effect |
|---|---|
| −f(x) | Reflects over the x-axis (flips vertically) |
| f(−x) | Reflects over the y-axis (flips horizontally) |
A negative sign in front of the entire function flips it vertically; a negative sign applied only inside the parentheses (on the x) flips it horizontally — where the negative sign sits changes everything.
Combining multiple transformations
The PSAT often combines two or more transformations in a single equation. Break the equation apart piece by piece, and identify each transformation independently.
Describe the transformation from f(x) to g(x) = 2f(x − 3) + 1.
Identify the horizontal shift (inside the parentheses) separately from everything happening outside the parentheses (vertical stretch/compression, reflection, vertical shift). Mixing these two groups together is a common source of confusion.
Common PSAT traps
- Reversing horizontal shift direction: f(x − h) shifts right for positive h, left for negative h — opposite of what the sign suggests at first glance.
- Confusing where the negative sign sits: −f(x) flips vertically; f(−x) flips horizontally.
- Applying stretch/compression to the wrong part: a coefficient outside the function affects the output (vertical); a coefficient on x inside affects the input (horizontal, and behaves oppositely from vertical stretch rules).
- Losing track of a transformation in a combined equation: isolate each piece of the equation one at a time rather than trying to visualize everything at once.
Test day strategy for function transformations
| Question signal | Fastest approach |
|---|---|
| Constant added/subtracted outside f(x) | Vertical shift — up for positive, down for negative |
| Constant added/subtracted inside the parentheses | Horizontal shift — opposite the sign shown |
| Coefficient multiplying the whole function | Vertical stretch (|a|>1) or compression (0<|a|<1) |
| Negative sign present | Check its position: outside flips vertically, inside (on x) flips horizontally |
| Multiple changes in one equation | Separate and identify each transformation independently |
Now put it to work
One comprehensive quiz covering the full topic — shifts, stretches, compressions, and reflections.
Transformations of Functions
Shifts, stretches, compressions, and reflections.