Function Transformation Calculator

Visualize how shifting, stretching, and reflecting affects a function's graph. Explore translations, dilations, and reflections for common parent functions like quadratic, absolute value, and sine.

Parent: f(x) = x²

Transformed

Grid View Range (Zoom)

250

Parameters

Parent Function

Vertical Stretch (a = 1)

-1010

Horizontal Stretch (b = 1)

-1010

Horizontal Shift (h = 0)

-2020

Vertical Shift (k = 0)

-2020

Transformed Equation

y = x²

Quick Note:

The standard form is y = a * f(b(x - h)) + k. Adjust the sliders to see how each constant moves or deforms the parent graph.

Decoding the Transformation

Inside the Function (h, b)

Affects the horizontal axis. These are often counter-intuitive: adding h shifts left, and a large b value compresses the graph.

Outside the Function (a, k)

Affects the vertical axis. These follow our intuition: adding k moves the graph up, and multiplying by a value > 1 stretches it taller.

How Function Transformations Work

Function transformation involves modifying a "parent" function to create a new, related function. Each constant added or multiplied in the equation affects the graph in a specific way.

1
Vertical Shifts (k): Adding k to the output moves the graph up (k > 0) or down (k < 0).
2
Horizontal Shifts (h): Subtracting h from the input (x-h) moves the graph right (h > 0) or left (h < 0).
3
Vertical Dilations (a): Multiplying the output by a stretches it vertically (|a| > 1) or compresses it (|a| < 1).
4
Reflections: A negative "a" reflects the graph across the x-axis. A negative "b" (inside input) reflects it across the y-axis.

Goal

y = a * f(b(x - h)) + k

This general form captures all transformations: a (vertical stretch/reflection), b (horizontal stretch/reflection), h (horizontal shift), and k (vertical shift).

Example

Parent Function: f(x) = x^2 Transformation: y = 2(x - 3)^2 + 1 1. Horizontal Shift: Right by 3 units (h=3) 2. Vertical Stretch: Factor of 2 (a=2) 3. Vertical Shift: Up by 1 unit (k=1) Result: A parabola with vertex at (3, 1), twice as steep as x^2.

Frequently Asked Questions

What is a parent function?

A parent function is the simplest form of a function family (e.g., f(x) = x² for quadratics). All other functions in the family can be created by applying transformations to this base form.

Why does (x-h) shift to the right?

To get the same output as the original f(x), the input (x-h) must be larger by h units to compensate. Thus, if h=3, you need x=3 to get the value f(0), effectively shifting the graph to the right.

How do I reflect a graph across the x-axis?

Multiply the entire function by -1 (y = -f(x)). This flips all y-coordinates to their opposite values, creating a mirror image across the horizontal axis.

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