Function Transformer

Math Algebra • Functions

Written by STEM Calculators Team Published December 13, 2025 Updated February 24, 2026

Choose a base function (or enter a custom \(f(x)\)), then adjust \(a,b,h,k\) in \(\;g(x)=a\,f(b(x-h))+k\). Drag the graph to pan and use the mouse wheel to zoom (hold Shift for vertical pan/zoom).

Base function: Custom \(f(x)\):

Transform parameters for \(g(x)=a\,f(b(x-h))+k\)

Vertical scale \(a\):

Horizontal scale \(b\):

Shift \(h\) (right if +):

Vertical shift \(k\):

Graph window

\(x_{\min}\): \(x_{\max}\): Resolution:

900

Options:

Show steps & summary Split view (before/after) Show probe point + coordinates

Ready

Probe

x=0 · f(x)=… · g(x)=…

Choose a base function and parameters, then press Calculate. The tool outputs the transformed equation \(g(x)=a\,f(b(x-h))+k\) and plots both functions.

Drag to pan. Mouse wheel to zoom (cursor-centered). Hold Shift to pan/zoom vertically. Double-click to reset view.

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4. Function Transformer — Theory

Most “graph transformations” can be organized into one standard expression: \[ g(x)=a\,f\!\big(b(x-h)\big)+k. \] The parameters \(a,b,h,k\) describe vertical/horizontal stretches, reflections, and shifts.

Key idea: changes outside the function (multiply/add) affect the y-values. Changes inside the function (replace \(x\)) affect the x-values.

1) The standard form

\[ g(x)=a\,f\!\big(b(x-h)\big)+k. \]

Parameter	What it does	Quick memory
\(h\)	Horizontal shift: \(x\mapsto x-h\) shifts the graph right by \(h\) (left if \(h<0\)).	“Inside opposite”: \(x-h\) → right.
\(b\)	Horizontal scale/reflect: \(x\mapsto b(x-h)\) compresses/stretch horizontally by factor \(\tfrac{1}{\|b\|}\). If \(b<0\), reflect across the y-axis.	Multiply inside → inverse effect on width.
\(a\)	Vertical scale/reflect: multiply outputs by \(a\). Scale by \(\|a\|\). If \(a<0\), reflect across the x-axis.	Outside multiply → same effect on height.
\(k\)	Vertical shift: add \(k\). Up if \(k>0\), down if \(k<0\).	Outside add → up/down.

2) Why “inside” is reversed

If you replace \(x\) by \(2x\), points must move toward the \(y\)-axis to keep the same input to \(f\). For example, if \(f(1)=3\), then to get \(f(2x)=3\) you need \(2x=1\Rightarrow x=\tfrac12\). So the point \((1,3)\) becomes \((\tfrac12,3)\): a horizontal compression.

3) Before/after mapping of points

Start from a point \((u,\,f(u))\) on the base graph. For the transformed graph, the same output occurs when \(u=b(x-h)\). Solve for \(x\):

\[ u=b(x-h)\quad\Rightarrow\quad x=\frac{u}{b}+h\quad(b\ne 0). \]

Then the y-value transforms as \(y=a\,f(u)+k\). So (when \(b\neq 0\)) a base point maps to:

\[ (u,\,f(u))\ \longmapsto\ \left(\frac{u}{b}+h,\ a\,f(u)+k\right). \]

4) Inverse transform idea

If you know \(g(x)=a\,f(b(x-h))+k\) and want to “undo” it (map points back to the base graph), you can reverse the steps:

\[ u=b(x-h),\qquad y_{\text{base}}=\frac{y-k}{a}\quad(a\ne 0). \]

If \(a=0\), then \(g\) collapses to a constant (when defined), and the transformation is not invertible.

5) Worked example

Base \(f(x)=x^2\). Apply: vertical stretch \(2\), shift right \(1\), reflect over the \(x\)-axis. That means \(a=-2,\ b=1,\ h=1,\ k=0\).

\[ g(x)= -2\,f(x-1) = -2(x-1)^2. \]

Frequently Asked Questions

What is a function transformation in algebra?

A function transformation changes a graph by shifting, stretching, compressing, or reflecting it. These changes are created by modifying the input x, the output f(x), or both.

How do a and k change the graph in y = a f(x) + k?

The value a scales the outputs vertically and reflects the graph across the x-axis when a is negative. The value k shifts the entire graph up when k > 0 and down when k < 0.

How do b and h change the graph in y = f(b(x - h))?

The value h shifts the graph right by h (left if h is negative). The value b scales horizontally, with b > 1 compressing the graph and 0 < b < 1 stretching it, and a negative b reflecting across the y-axis.

Why does horizontal scaling feel reversed compared to vertical scaling?

Horizontal changes act on the input x rather than the output, so replacing x with b x changes where the same output occurs. This is why larger b values compress the graph horizontally instead of stretching it.

When does a transformation change the domain or range?

Shifts and scalings can move the set of outputs and can also move or resize the interval of x-values you are considering. Additional domain restrictions still apply whenever the original function has constraints, such as division by zero, sqrt inputs, or log inputs.

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