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Transformations of Graphs

Master transformations of graphs with interactive lessons and practice problems! Designed for students like you!

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Transformations of Graphs

Study Guide

Key Definition

Transformations of graphs involve translations, rotations, reflections, and dilations. Each transformation has unique properties like preserving size and shape or only preserving shape.

Important Notes

Translations move the graph without rotating or flipping it: $(x, y) \rightarrow (x - 5, y + 3)$
Rotations turn the graph around a point: $(x, y) \rightarrow (y, -x)$
Reflections flip the graph over a line. For reflection across the line y = x + 2: $(x, y) \rightarrow (y - 2, x + 2)$
Dilations change the size but not the shape of the graph
Understanding these transformations helps in graphing functions

Mathematical Notation

Notation specific to each transformation is shown with its formula above.

Why It Works

Transformations are based on applying consistent rules to every point on a graph, ensuring each transformation maintains specific properties like shape or size.

Remember

Use transformations to easily predict how graphs change when equations are modified.

Quick Reference

Translation Formula:$(x, y) \rightarrow (x + a, y + b)$

Rotation Formula (90° Clockwise):$(x, y) \rightarrow (y, -x)$

Reflection Formula (across line y = x + 2):$(x, y) \rightarrow (y - 2, x + 2)$

Dilation Formula:$(x, y) \rightarrow (kx, ky)$

Understanding Transformations of Graphs

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A translation shifts every point on a graph by the same amount horizontally and vertically. For example, $(x, y) \rightarrow (x + a, y + b)$ moves the graph a units to the right and b units up.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the resulting point when $(x, y)$ is translated by $(x - 5, y + 3)$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A designer is rotating a square design 90^\\circ clockwise on a digital canvas. What is the new position of the point located at $(3, 4)$?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider a shape reflected across the line $y = x + 2$. What happens to a point located at $(3, 5)$?

Challenge Quiz

Single Choice Quiz

Advanced

After a dilation with a scale factor of $2$, what is the size of a triangle with original side lengths of $3$, $4$, and $5$?

Recap

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Review key concepts and takeaways