PSAT/NMSQT Math: Transformations of Functions
Once you know the graph of a basic function, transformations let you predict the graph of a related function without plotting a single new point. This free, complete lesson covers vertical and horizontal shifts, reflections over the x-axis and y-axis, stretches and compressions, and applying transformations to specific points and real-world contexts. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quiz below.
1. Vertical Shifts
Adding a constant k outside a function, g(x) = f(x) + k, shifts the entire graph vertically: up if k > 0, down if k < 0. Every y-value on the original graph simply increases or decreases by k, while x-values stay the same.
Worked Example: Which function represents f(x) = x² shifted up 3 units?
g(x) = x² + 3
1. Which function represents f(x) = x² shifted up 3 units? (see worked example above)
Explanation: Adding a constant outside the function shifts the graph vertically; +3 shifts it up 3 units.
2. If a point (2, 5) lies on the graph of f(x), and g(x) = f(x) + 4, what point lies on the graph of g?
Explanation: Adding 4 outside the function increases every y-value by 4, so (2, 5) becomes (2, 5 + 4) = (2, 9).
3. A function h(x) = f(x) − 7 is graphed. Compared to f(x), how is h(x)'s graph positioned?
Explanation: Subtracting outside the function shifts the entire graph down, here by 7 units.
2. Horizontal Shifts
Adding or subtracting a constant h inside the function's input, g(x) = f(x − h), shifts the graph horizontally. This is counterintuitive: f(x − h) shifts right by h, while f(x + h) shifts left by h, the opposite direction from what the sign might suggest.
Worked Example: How does y = (x + 4)² compare to y = x²?
Adding inside the parentheses shifts the graph left by 4 units
1. How does y = (x + 4)² compare to y = x²? (see worked example above)
Explanation: A change inside the parentheses shifts the graph horizontally; (x + 4) shifts it left, opposite to the sign shown.
2. What transformation takes f(x) to g(x) = f(x − 6)?
Explanation: f(x − h) always shifts the graph right by h, so f(x − 6) shifts right 6 units.
3. If a point (3, 2) lies on the graph of f(x), and g(x) = f(x − 5), what point lies on the graph of g?
Explanation: Shifting right 5 units moves every x-value up by 5: (3 + 5, 2) = (8, 2).
3. Reflections Over the X-Axis and Y-Axis
Multiplying the entire function by −1, g(x) = −f(x), reflects the graph over the x-axis (flipping it vertically). Replacing x with −x, g(x) = f(−x), reflects the graph over the y-axis (flipping it horizontally).
Worked Example: Find the equation of f(x) = x³ reflected over the x-axis.
g(x) = −f(x) = −x³
1. Find the equation of f(x) = x³ reflected over the x-axis. (see worked example above)
Explanation: A reflection over the x-axis flips the output's sign: g(x) = −f(x) = −x³.
2. Identify the reflection of y = f(x) = ex over the y-axis.
Explanation: A reflection over the y-axis replaces x with −x throughout the function: f(−x) = e−x.
3. A graph of y = √x is reflected over the x-axis. What is the equation of the new graph?
Explanation: Reflecting over the x-axis negates the entire output: g(x) = −f(x) = −√x.
4. Stretches, Compressions, and Real-World Context
Multiplying the output by a constant a > 1 (g(x) = a · f(x)) stretches the graph vertically; 0 < a < 1 compresses it. Multiplying the input by b (g(x) = f(bx)) with |b| > 1 compresses the graph horizontally; 0 < |b| < 1 stretches it horizontally. In real-world models, a transformation often corresponds to a change described directly in the problem, such as a shifted baseline or a scaled rate.
Worked Example: Which equation represents a horizontal compression by a factor of ½ for f(x) = x²?
g(x) = f(2x) = (2x)² = 4x²
1. Which equation represents a horizontal compression by a factor of ½ for f(x) = x²? (see worked example above)
Explanation: Compressing horizontally by a factor of ½ means replacing x with 2x inside the function, giving g(x) = (2x)².
2. A sinusoidal model of daily temperature has an average value of 60°F. If the average shifts to 70°F while the seasonal pattern stays otherwise identical, what kind of transformation occurred?
Explanation: Raising the entire baseline average by a fixed amount, without changing the shape or timing of the seasonal swings, is a vertical shift, here up by 10°F.
3. If g(x) = 3f(x) and f(2) = 4, what is g(2)?
Explanation: A vertical stretch multiplies the output by the constant: g(2) = 3 · f(2) = 3 · 4 = 12.
Common Mistakes to Avoid
- Assuming a horizontal shift moves in the same direction as the sign inside the parentheses. f(x − h) shifts right, and f(x + h) shifts left, the opposite of what the sign suggests.
- Confusing where a constant goes for a vertical shift versus a horizontal shift. A constant added or subtracted outside the function shifts vertically; inside the function's input, it shifts horizontally.
- Mixing up which reflection flips over which axis. −f(x) reflects over the x-axis (flips outputs); f(−x) reflects over the y-axis (flips inputs).
- Confusing a horizontal stretch/compression factor with its visual effect. Multiplying x by a number greater than 1 compresses the graph horizontally, not stretches it, since points reach the same output faster.
- Forgetting to apply a transformation to every coordinate of a known point. A vertical transformation changes only the y-coordinate; a horizontal transformation changes only the x-coordinate.
Frequently Asked Questions
Why does f(x − h) shift right instead of left?
Think of it as asking "what input to f gives the same output as f(x) did at x = 0 in the original?" Since you need a larger x (specifically x = h larger) to get the same result, the whole graph appears to move to the right.
How do I combine multiple transformations in the correct order?
A common safe order is: horizontal shifts and reflections first (inside the function), then vertical stretches/compressions and reflections, and finally vertical shifts (outside the function), matching the order operations would be applied to a specific x-value.
Where can I practice more problems like these?
The Transformations of Functions quiz in the PSAT/NMSQT Math Question Bank includes additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT/NMSQT. You can also browse the full PSAT/NMSQT Math Question Bank.