Function Transformations
Explore function transformations with interactive sliders. Shift, reflect, stretch, and compress a function while comparing the original and transformed graphs.
Choose a parent function
Transform the function
g(x) = a·f(b(x - h)) + k
- h → horizontal shift
- k → vertical shift
- a → vertical stretch/compression and reflection
- b → horizontal stretch/compression and reflection
h = 3 → right 3 · h = -3 → left 3
k = 4 → up 4 · k = -4 → down 4
Preset transformations
g(x)
x [-10, 10]
Graph settings
Display and accessibility
Grid settings
Polar draws radial rings and 30° spokes. Type r = 4*sin(3*theta) to plot a polar curve.
X-axis settings
Y-axis settings
More options
Equal scaling keeps circles round. Independent scaling stretches each axis to fill the view.
Angle mode
Degrees: sin(90) = 1. Applies to graphs, roots, and intersections.
Drag to pan · scroll or pinch to zoom · keyboard: arrows pan, + / - zoom, 0 reset
Current transformation
Before / after comparison
Example
What are function transformations?
A function transformation takes a familiar parent curve and moves, flips, or reshapes it: shifts slide it, reflections mirror it, and stretches squeeze it. If you learn best by watching a graph respond, pick a parent, drag a slider, and see the equation change with the curve.
Try an example
Remember: horizontal transformations work from inside the function. f(x - h) moves the graph right, while f(bx) compresses it horizontally when |b| > 1.
Transformation rules
| Right h | f(x - h) |
| Left h | f(x + h) |
| Up k | f(x) + k |
| Down k | f(x) - k |
| Reflect over x-axis | -f(x) |
| Reflect over y-axis | f(-x) |
| Vertical stretch | a f(x), |a| > 1 |
| Vertical compression | a f(x), 0 < |a| < 1 |
| Horizontal compression | f(bx), |b| > 1 |
| Horizontal stretch | f(bx), 0 < |b| < 1 |
Transformations FAQ
What are function transformations?
Function transformations change the position, orientation, or shape of a parent function to create a new graph.
What does f(x - h) do?
It shifts the graph horizontally to the right by h units when h is positive.
What does f(x) + k do?
It shifts the graph vertically up by k units when k is positive.
What does a negative sign outside a function do?
-f(x) reflects the graph across the x-axis.
What does f(-x) do?
It reflects the graph across the y-axis.
What is the difference between a horizontal and vertical stretch?
A vertical stretch changes the output values using a multiplier outside the function. A horizontal stretch changes the input using a multiplier inside the function.
Why does f(2x) compress a graph?
Because the input reaches the same function value with half the x-distance. Therefore the graph is horizontally compressed by a factor of 2.