Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for algebra 2 function transformations? Then this Algebra 2 lesson on Transformations of Functions and Their Graphs explains each idea step by step, with worked examples.
Quick answer: Transformations change a parent graph predictably: f(x) + k shifts up, f(x − h) shifts right, −f(x) reflects across the x-axis, af(x) stretches vertically, and f(bx) compresses horizontally. Even functions are symmetric about the y-axis; odd functions about the origin.
Standards: F-BF.B.3, F-IF.C.7b · Level: Grade 11 Algebra 2
How do shifts and reflections change a graph?
Starting from a parent function y = f(x), each change to the equation moves or flips the graph in a predictable way. Changes inside the parentheses act on x and work “backwards”; changes outside act on y.
| Equation | Effect on the graph of f |
|---|---|
| y = f(x) + k | Shift up k (down if k < 0) |
| y = f(x − h) | Shift right h (left if h < 0) |
| y = −f(x) | Reflect across the x-axis |
| y = f(−x) | Reflect across the y-axis |
| y = af(x) | Stretch vertically by a if |a| > 1; shrink if 0 < |a| < 1 |
| y = f(bx) | Compress horizontally by 1/b if |b| > 1; stretch if 0 < |b| < 1 |
What are the key parent functions?
| Parent | Rule | Domain | Range |
|---|---|---|---|
| Quadratic | f(x) = x2 | All reals | y ≥ 0 |
| Absolute value | f(x) = |x| | All reals | y ≥ 0 |
| Square root | f(x) = √x | x ≥ 0 | y ≥ 0 |
| Cube root | f(x) = ∛x | All reals | All reals |
| Exponential | f(x) = 2x | All reals | y > 0 |
- Identify the parent function.
- Locate the key point (vertex, start point, center, or asymptote) of the parent.
- Apply stretches and reflections, then the shifts.
- Move the key point and plot a few more points.
Graph g(x) = 2√(x − 2) + 1. The parent √x starts at (0, 0). The shift moves the start to (2, 1). The stretch by 2 turns the point (1, 1) of the parent into (3, 3) and (4, 2) into (6, 5).
What are even and odd functions?
A function is even if f(−x) = f(x) for every x; its graph is symmetric about the y-axis. It is odd if f(−x) = −f(x); its graph is symmetric about the origin.
Horizontal changes work opposite to what the sign suggests: f(x + 3) moves the graph left 3, not right. Also, a vertical stretch multiplies the y-values, so the key point moves up or down, not sideways.
Track one key point: apply the horizontal changes to its x-coordinate and the vertical changes to its y-coordinate.
- Outside changes act on y: f(x) + k, af(x), and −f(x).
- Inside changes act on x and work backwards: f(x − h) shifts right h.
- A key point of the parent, such as a vertex or start point, shows the move in one step.
- Square root and cube root graphs start at or are centered on the shifted key point.
- Even: f(−x) = f(x). Odd: f(−x) = −f(x).
Transformations of Functions and Their Graphs: frequently asked questions
Why does f(x − 3) shift the graph to the right?
The output of f(x − 3) at x = 5 equals f(2), which the parent produces at x = 2. So every point of the parent moves 3 units right to produce the new graph.
What is the difference between f(−x) and −f(x)?
−f(x) changes the sign of the output, reflecting the graph across the x-axis. f(−x) changes the sign of the input, reflecting the graph across the y-axis.
How do you find the domain of a transformed square root function?
Set the expression under the root greater than or equal to 0. For y = √(x + 7) − 2, x + 7 ≥ 0 gives x ≥ −7.
How can you tell if a function is even or odd from its equation?
Replace x with −x and simplify. If you get the original function it is even; if you get its opposite it is odd; otherwise it is neither.
Does the order of transformations matter?
Yes, a stretch or reflection should be applied before a vertical shift. For y = 2f(x) + 3, double the outputs first and then add 3, which gives a different graph from adding 3 first and then doubling.
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