Math · College algebra · Concept
Function transformations and their graphs
A transformation moves or reshapes a graph without changing its basic shape. In y = a·f(b(x − h)) + k, h shifts the graph right by h, k shifts it up by k, a stretches it vertically by |a| and flips it if a < 0, and b scales horizontal distances by 1/|b| and flips the graph if b < 0.
Shifts move the whole graph
Adding k outside the function moves every point up by k, or down if k < 0. Replacing x with x − h moves every point right by h, or left if h < 0. The horizontal shift feels backward: y = f(x − 3) moves right, because the input must be 3 larger to give the same output.
Stretches and reflections
Multiplying the output by a stretches the graph vertically by a factor of |a|; a negative a also reflects it across the x-axis. Multiplying the input by b scales horizontal distances by 1/|b|, so b = 2 squeezes the graph toward the y-axis; a negative b reflects it across the y-axis. An even function such as x² looks the same after that reflection.
The four changes at a glance
Each change acts on one direction only.
| Change | Effect on the graph |
|---|---|
| f(x) + k | Up k (down if k < 0) |
| f(x − h) | Right h (left if h < 0) |
| a·f(x) | Vertical stretch by |a|; reflect across the x-axis if a < 0 |
| f(bx) | Horizontal distances times 1/|b|; reflect across the y-axis if b < 0 |
Order matters
Work from the inside out: the horizontal changes act on x first, then a stretches the outputs, then k shifts them. For y = 2f(x − 3) + 1, shift right 3, stretch vertically by 2, then shift up 1. Shifting up before stretching would double the shift too.
Factor b before reading the shift
In y = f(2x − 6), the shift is not 6. Factor the input first: 2x − 6 = 2(x − 3), so the graph is squeezed horizontally by a factor of 2 and shifted right 3.
Track a key point
A point (x, y) on y = f(x) moves to (x/b + h, a·y + k) on y = a·f(b(x − h)) + k. Following the vertex or an intercept is the quickest check of a transformed graph.
Common mistakes
- Shifting the wrong way: y = f(x − 3) moves the graph right, not left.
- Reading f(2x) as a horizontal stretch: it halves horizontal distances.
- Applying the vertical shift before the stretch: 2(f(x) + 1) is not 2f(x) + 1.
- Reading the shift before factoring out b: f(2x − 6) = f(2(x − 3)) shifts right 3, not 6.
Key terms
- Function transformation
- A change to a function’s input or output that shifts, reflects or scales its graph. Changes inside the input act horizontally; changes outside act vertically.
- Function
- A rule assigning exactly one output to each allowed input. The notation f(x) means the output of f at x; it does not mean f multiplied by x.
- Even function
- A function with f(−x) = f(x) for every x in its domain, such as x² or cos x. Its graph is a mirror image across the y-axis.
- Odd function
- A function with f(−x) = −f(x) for every x in its domain, such as x³ or sin x. Its graph looks the same after a half-turn about the origin.
- Domain
- The set of inputs for which a function or expression is defined. In the real numbers that rules out zero denominators, negative numbers under even roots and inputs of logarithms that aren’t positive.
Work through an example
Describe how to get the graph of g(x) = 2(x − 3)² + 1 from the graph of f(x) = x², and find where the vertex and the points (±1, 1) end up.
Transform the graph of y = x² →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.
Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
Open Function Transformations Check the points in Math Open worked example on a board Transformations in Math ReferenceYour existing work stays on this device. Examples open as editable copies.