Function Transformation Grapher

About the Function Transformation Grapher

Explore translations, stretches and reflections by editing a base function.

How to use this calculator

1. Choose a base function, such as \(f(x)=x^2\) or \(f(x)=|x|\).
2. Edit the transformation settings to move the graph left, right, up, or down.
3. Add stretches, compressions, or reflections to change the graph’s shape and orientation.
4. Compare the original and transformed graphs, and use the displayed formula to check your result.

The formula explained

A transformed function often has the form \(g(x)=a\,f(b(x-h))+k\). This formula tells you how the base graph changes: horizontal shifts, horizontal stretches or shrinks, vertical stretches or shrinks, and reflections.

• x = the input value on the horizontal axis
• f(x) = the original base function
• g(x) = the transformed function
• a = vertical stretch or shrink factor, and reflection across the x-axis if negative
• b = horizontal stretch or shrink factor, and reflection across the y-axis if negative
• h = horizontal shift, right if positive and left if negative
• k = vertical shift, up if positive and down if negative

Step by step method

1. Start with the base graph, for example \(f(x)=x^2\).
2. Apply shifts first, then stretches or reflections, using the transformation parameters you chose.
3. Write the new rule in the form \(g(x)=a\,f(b(x-h))+k\), then check a few points to confirm the graph matches.

Worked example

Problem. Transform the base function \(f(x)=x^2\) by shifting it right 2 units, up 3 units, and reflecting it across the x-axis.

1. Right 2 units means replace \(x\) with \(x-2\), so the graph becomes \((x-2)^2\).
2. Up 3 units means add 3, and reflecting across the x-axis means multiply by \(-1\), so the rule becomes \(g(x)=-(x-2)^2+3\).
3. The transformed function is \(g(x)=-(x-2)^2+3\).

Tips and common mistakes

• Inside changes, like \(x-h\) or \(b(x-h)\), affect the graph horizontally, and outside changes, like \(a\) and \(+k\), affect it vertically.
• A common mistake is mixing up left and right, remember \(x-2\) shifts right 2, while \(x+2\) shifts left 2.

Frequently asked questions