mathematicsfunctionstransformationsactive lesson

Active Lesson: Function Transformations and Inverses

Mathimatikos TeamJune 8, 20268 min

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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.

Interaktiv graf

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

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