Functions and Transformations
Once you understand what a function is, the next step is understanding how functions change when you modify them. Transformations let you take any function and shift, flip, or stretch it , and recognizing these patterns makes graphing dramatically faster.
In this lesson
1 Quick Review: Functions and Graphs
A function f(x) maps each input x to exactly one output. Its graph is the set of all points (x, f(x)). The parent function is the simplest version , f(x) = x², f(x) = |x|, f(x) = √x. Every transformation modifies a parent function in a predictable way.
Instead of memorizing dozens of different graphs, you can learn a handful of parent functions and the transformation rules. Any graph is a transformed version of something simpler.
2 Vertical Shifts and Stretches
Vertical shift: f(x) + k shifts the graph up by k units (down if k is negative). Every y-value increases by k. The shape is unchanged.
Vertical stretch/compression: a·f(x) multiplies every y-value by a. If |a| > 1, the graph stretches (taller). If 0 < |a| < 1, it compresses (shorter).
3 Horizontal Shifts and Stretches
Horizontal shift: f(x − h) shifts the graph right by h units (left if h is negative). This is counterintuitive , subtracting h moves right.
Horizontal stretch/compression: f(bx) compresses horizontally by factor b (if b > 1) or stretches (if 0 < b < 1).
f(x − 3) shifts RIGHT, not left. Think of it this way: to get the same output you used to get at x=0, you now need x=3. So the whole graph moves right.
4 Reflections
−f(x): reflects over the x-axis (flips upside down , every y-value negated).
f(−x): reflects over the y-axis (flips left-right , every x-value negated).
5 Inverse Functions
The inverse function f⁻¹(x) reverses what f does. If f(2) = 7, then f⁻¹(7) = 2. Graphically, f⁻¹ is the reflection of f over the line y = x (swap x and y coordinates of every point).
To find the inverse algebraically: replace f(x) with y, swap x and y, then solve for y.
A function only has an inverse if it is one-to-one , each output corresponds to exactly one input (passes the horizontal line test). y = x² fails the horizontal line test, so it doesn't have an inverse over all reals. That's why √x is only defined for x ≥ 0.
Practice Problems
Sources & Further Reading
The explanations on this page draw on the following established sources. We link to primary and secondary sources so you can verify claims and go deeper on any topic.