mathematicsfunctionstransformationsactive lesson

Active Lesson: Function Transformations and Inverses

Mathimatikos TeamJune 8, 20268 Min

Einige ergänzende Texte in diesem Bereich sind derzeit auf Englisch verfügbar.

Auf dieser Seite

Before You Start

Function notation
Basic graph reading
Substitution into formulas

Lesson time

18 min

Objectives

What You Will Learn

Describe how $A$ , $B$ , $C$ , and $D$ change the graph of $F (x) = A f (B (x - C)) + D$ .
Predict a transformation before drawing the graph.
Connect transformations with composition and inverse functions.

Activity

Preview Activity: Move a Parent Function

Start with the parent function $f (x) = x^{2}$ . Instead of memorizing rules, change one number at a time and describe what you see.

Compare $f (x) = x^{2}$ with $g (x) = 2 x^{2}$ . What happens to each $y$ -value?
Compare $f (x) = x^{2}$ with $h (x) = (x - 3)^{2}$ . Which direction does the graph move?
Compare $f (x) = x^{2}$ with $k (x) = - (x + 1)^{2} + 2$ . List the transformations in the order they affect the graph.

Hint

Choose one input, such as $x = 2$ , and compare the output before and after the change.

Solution

$g (x)$ stretches the graph vertically by a factor of $2$ .

$h (x)$ shifts the graph $3$ units to the right.

$k (x)$ shifts left by $1$ , reflects across the $x$ -axis, and shifts up by $2$ .

Definition

The Transformation Template

F (x) = A f (B (x - C)) + D

$A$ changes the output of the parent function, so it creates a vertical stretch or reflection.

$B$ changes the input before the parent function acts, so it creates a horizontal stretch or reflection.

$C$ moves the graph horizontally and $D$ moves it vertically.

Visual Comparison

The transformed graph $y = 2 (x - 1)^{2} + 1$ is the parent parabola shifted right, stretched vertically, and moved up.

Interaktive Grafik

Why Horizontal Changes Feel Backwards

In $f (x - 3)$ , the graph moves right because the input must be $3$ larger before the parent function receives the same value.

For example, $f (x - 3)$ behaves like $f (0)$ when $x = 3$ . That is why the old point at $x = 0$ appears at $x = 3$ .

Worked example

Worked Example: Read a Formula

Describe the transformations in $g (x) = - 3 f (2 (x + 4)) - 5$ .

Solution

The graph shifts left $4$ units, compresses horizontally by a factor of $2$ , reflects across the $x$ -axis, stretches vertically by $3$ , and shifts down $5$ units.

Definition

Inverse Functions

f (f^{- 1} (x)) = x and f^{- 1} (f (x)) = x

An inverse function undoes the action of the original function.

Graphically, the inverse is reflected across the line $y = x$ . If the reflected graph fails the vertical line test, the original function needs a domain restriction.

Summary

Transformations are easier when you separate input changes from output changes.
Changes outside $f$ affect $y$ -values; changes inside $f$ affect $x$ -values.
An inverse reverses a function, but it only exists when each output comes from one input.

Practice

Try These

Describe the graph of $y = 4 (x - 2)^{2} - 7$ from the parent function $y = x^{2}$ .
Find the inverse of $f (x) = 3 x - 8$ .
Explain why $f (x) = x^{2}$ has no inverse unless its domain is restricted.

Practice functions in the Greek quiz