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Transformation of Function

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Transformation of Function

Study Guide

Key Definition

A transformation is a manipulation of the shape or position of a function on a Cartesian plane. For example, a horizontal shift of $y = x^2$ results in $y = (x - h)^2$.

Important Notes

Horizontal shifts involve changing the x-value by $h$.
Vertical shifts change the y-value by $k$.
A function $y = f(x)$ can be transformed to $y = f(x - h) + k$.
Reflections across the x-axis change $y = f(x)$ to $y = -f(x)$.
Scaling alters the steepness of the graph.

Mathematical Notation

$(x - h)$: shift right by h units

$(x + h)$: shift left by h units

$f(x) + k$: shift up by k units

$f(x) - k$: shift down by k units

$a·f(x)$: vertical stretch by factor a (|a|>1) or compression (0<a<1)

$f(bx)$: horizontal compression by factor 1/b (b>1) or stretch (0<b<1)

$-f(x)$: reflection over x-axis

$f(-x)$: reflection over y-axis

Remember to use proper notation when solving problems

Why It Works

Transformations allow us to understand how functions behave under different conditions using $y = f(x)$ and its transformations.

Remember

Transformations can shift, reflect, and scale a graph, impacting its shape and position on the plane.

Quick Reference

Horizontal Shift:$y = f(x - h)$

Vertical Shift:$y = f(x) + k$

Reflection:$y = -f(x)$

Understanding Transformation of Function

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A horizontal shift moves every point of y = x^2 left or right. For example, y = (x - 3)^2 shifts the graph 3 units to the right. (See the reference graph in the assets.)

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the transformation of $y = x^2$ when shifted 3 units to the right?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine adjusting the graph of a skateboard ramp described by $y = x^2$ by moving it 2 units up.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Describe the transformation of $y = -x^2 + 4$.

Challenge Quiz

Single Choice Quiz

Advanced

What is the transformation of $y = x^2$ to $y = -2(x + 1)^2 - 3$?

Recap

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Review key concepts and takeaways