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Mathematics

Function Transformations

Coordinate plane

Base function

Family being explored

g(x)=a·(b(x−h))²+k

Vertical change · a

Horizontal change · b

Horizontal translation · h

Vertical translation · k

Drag the background or use the zoom buttons to change the view.

Display layers

Graph of fCharacteristic pointsGuidesRead coordinates

f(x)=x²No transformation applied

Domain: ℝ
Range: [0; +∞)

Main points

Sample point(−1; 1)
Vertex · common intersection with Ox and Oy(0; 0)
Sample point(1; 1)

Number lines and table of values

DomainD = ℝ

RangeT = [0; +∞)

Equivalent value table

A few finite values of the base function
x	−6	−4	−2	0	2	4	6
f(x)	36	16	4	0	4	16	36

x =

Hover near a curve, tap it, or use the ← → keys on the graph.

Choose a parameter or manipulate the graph directly to start exploring.

Symmetry

Parity of f: even function

What the parameters mean

a: |a| stretches/compresses vertically; negative a reflects across Ox.
b: |b| acts horizontally by the reciprocal; negative b reflects across Oy.
h: Positive h in f(x − h) shifts the graph to the right.
k: Positive k shifts the graph upward.

Step 0.25; a and b must be nonzero. Values snap to the nearest step within the slider bounds.

Model notes

a acts vertically; b acts horizontally with the factor 1/|b|; h and k translate the graph horizontally and vertically respectively.
b(x − h) is one complete group and must not be read as bx − h.
For an even function, the reflection across Oy may leave the graph unchanged; for an odd function, the reflections across Ox and across Oy may give the same graph.
For f(x) = x, the graph only reveals the product a·b, so the two parameters cannot be recovered separately.
For 2ˣ and log₂ x, a translation can give the same graph as a stretch (for example 2·2ˣ = 2ˣ⁺¹ and log₂(2x) = log₂ x + 1), so the graph does not reveal each parameter separately.
sin x and cos x use radians; the graph of g has period 2π/|b| and amplitude |a|.
Free exploration and Step-by-step transformations offer ten base functions: x, x², x³, |x|, √x, 1/x, 2ˣ, log₂ x, sin x, cos x. The Predict the graph mode uses the five base functions x, x², |x|, √x, 1/x.
Both axes use the same unit length, so y = x rises at 45°. Curves are polylines through points sampled within the view; a branch approaching an asymptote is drawn to the edge of the frame.
Free exploration also accepts entered functions of x. Entered functions are sampled numerically within the view, without full domain, range or asymptote analysis; very small details or rapid oscillations may be missed. In the input field, ln is the base-e logarithm and log the base-10 logarithm. Derivatives and 3D graphs are not included.

Library

Mathematics

Function Transformations

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Resource information and sources

Updated in the library: 1 October 2026

About the library and usage rights

Function Transformations

Coordinate plane

Function

Parameters a, b, h, k

Drag on the graph

Result

Read coordinates

Explanation

Function Transformations

Coordinate plane

Function

Parameters a, b, h, k

Drag on the graph

Result

Read coordinates

Explanation