Topic 1.12B · CED: Transformations of Functions

Dilations of Functions

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01
$g(x) = a \cdot f(x)$ : $(x, y)$ on $f$ becomes ( $x$ , ay) on $g$ — bigger |a| stretches, smaller |a| compresses.
02
$g(x)$ = f(bx): $(x, y)$ on $f$ becomes (x/b, y) on $g$ — bigger |b| compresses, smaller |b| stretches. This is the opposite of the vertical case.
03
$g(x) = f(-x)$ : $(x, y)$ on $f$ becomes $(-x, y)$ on $g$ — reflects over the y-axis. Compare to $g(x) = -f(x)$ , which reflects over the x-axis.

Practice 24 problems Next: Chapter 1.13 Function Model Selection and Assumption Articulation

Multiplicative Transformations: The Big Picture

CONCEPT

Two Kinds of Dilations

$g(x) = a \cdot f(x)$ → VERTICAL dilation (stretches/compresses up-down) — matches intuition.

$g(x)$ = f(bx) → HORIZONTAL dilation (stretches/compresses left-right) — the effect is the OPPOSITE of what feels natural.

We'll reuse the SAME generic function $f(x)$ from 1.12A, with the same two tracking points, so you can compare shifting vs. dilating side by side.

Vertical Dilation: g(x) = a·f(x)

Multiplying the OUTSIDE of the function scales every y-value by a. If |a| > 1, points move farther from the x-axis (stretch). If 0 < |a| < 1, points move closer to the x-axis (compress).

Multiplying $f(x)$ by 2 doubles every y-value — points move twice as far from the x-axis.

Multiplying by 0.5 halves every y-value — points move halfway toward the x-axis.

KEY RULE

$g(x) = a \cdot f(x)$ : $(x, y)$ on $f$ becomes ( $x$ , ay) on $g$ — bigger |a| stretches, smaller |a| compresses.

Horizontal Dilation: g(x) = f(bx)

Here's the surprising one: f(2x) COMPRESSES horizontally (squeezes toward the y-axis), even though 2 seems like it should stretch things. Why? To reach the same output $f$ used to give at some input, $g$ now only needs an input HALF as large — so every feature happens closer to the y-axis, not farther.

f(2x) — every x-value is cut in HALF, squeezing the whole graph toward the y-axis.

f(0.5x) — every x-value DOUBLES, stretching the graph away from the y-axis.

KEY RULE

$g(x)$ = f(bx): $(x, y)$ on $f$ becomes (x/b, y) on $g$ — bigger |b| compresses, smaller |b| stretches. This is the OPPOSITE of the vertical case.

CONCEPT

The AP-Preferred Notation

The AP exam usually writes a horizontal stretch by factor $k$ as:

g(x) = f\left(\frac{x}{k}\right)

This is the SAME thing as f(bx) with $b = 1/k$ — just flipped into a fraction. $k>1$ stretches (matches $b<1$ in our notation); 0< $k<1$ compresses (matches $b>1$ ).

COMMON MISTAKE

The single most common error: assuming f(bx) with $b>1$ stretches the graph. It actually COMPRESSES it — bigger $b$ squeezes the graph inward.

When $b$ is a fraction (like 0.5), the horizontal stretch factor is its RECIPROCAL (here, stretched by 2, not by 0.5).

A negative $a$ or $b$ adds a reflection on top of the stretch/compression — handle the sign and the magnitude as two separate effects (see Section 4).

Quick check

$(6, -4)$ is on $f$ . Where is it on $g(x) = \frac{1}{2} f(3x)$ ?

Reflection Over the y-Axis: g(x) = f(−x)

In 1.12A, $g(x) = -f(x)$ flipped the graph over the x-axis (a NEGATIVE sign OUTSIDE the function). Here's its horizontal twin: $g(x) = f(-x)$ flips the graph over the y-axis (a negative sign INSIDE, attached to x).

Each point's x-coordinate flips sign — a point to the right of the y-axis lands the same distance to the left, and vice versa.

KEY RULE

$g(x) = f(-x)$ : $(x, y)$ on $f$ becomes $(-x, y)$ on $g$ — reflects over the y-AXIS. Compare to $g(x) = -f(x)$ , which reflects over the x-axis.

Worked Example

Worked example

Given $f(x) = x^{2} - 4x$ , find $g(x) = f(-x)$ .

01
Replace every $x$ with $(-x)$ :
$g(x) = f(-x) = (-x)^{2} - 4(-x) = x^{2} + 4x$
02
Notice the −4x term flipped sign (became +4x) while the $x^{2}$ term didn't change — squaring a negative gives back a positive.

COMMON MISTAKE

Don't confuse $g(x)=f(-x)$ (reflects over the y-axis, negative INSIDE) with $g(x)=-f(x)$ (reflects over the x-axis, negative OUTSIDE) — they do very different things.

When substituting $(-x)$ , apply the negative sign to EVERY occurrence of $x$ , including inside exponents — $(-x)^{2} = x^{2}$ , but $(-x)^{3} = -x^{3}$ .

Quick check

Which reflects over the $y$ -axis: $g(x) = f(-x)$ or $g(x) = -f(x)$ ?

Describing a Full Sequence of Transformations

AP-style questions often ask you to describe EVERY transformation packed into one function, in order. Here's how to read them off systematically.

Worked example

Describe, in order, all the transformations that turn $f(x)$ into $g(x) = -2f(3(x+1)) - 4$ .

CONCEPT

Reading Off Each Piece

Inside the parentheses, $(x+1)$ : horizontal shift LEFT 1.

The 3 multiplying $(x+1)$ : horizontal COMPRESSION by factor 3 (equivalently, f(x/k) form with $k=1/3$ ).

The −2 out front: reflection over the x-axis, AND a vertical stretch by factor 2.

The −4 at the very end: vertical shift DOWN 4.

Standard AP phrasing, in order: "Shift left 1, compress horizontally by a factor of 3, reflect over the x-axis and stretch vertically by a factor of 2, then shift down 4."

Step-by-Step: Dilating Algebraically

Worked example

Given $f(x) = x^{2} + 2x - 3$ , find $g(x) = 3f(x)$ , and separately $g(x) = f(3x)$ .

Vertical dilation: g(x) = 3f(x)

01
Multiply the ENTIRE function by 3:

g(x) = 3f(x) = 3x^{2} + 6x - 9

Horizontal dilation: g(x) = f(3x)

01
Replace every $x$ in $f(x)$ with (3x):

g(x) = f(3x) = (3x)^{2} + 2(3x) - 3

02
Expand and simplify:

g(x) = 9x^{2} + 6x - 3

Step-by-Step: Dilating Numerically

Worked example

Given the table of values for $f$ below, let $g(x) = 2f(x - 1)$ . Find $g(3)$ .

x	f(x)
0	5
1	8
2	2
3	6

01
$g(3) = 2 \cdot f(3-1) = 2 \cdot f(2)$ — first shift the input, then apply the vertical stretch.
02
From the table, $f(2) = 2$ .
03
$g(3) = 2 \cdot 2 = 4$ .

Domain and Range Under Dilation

CONCEPT

The Rule

Vertical dilation $g(x)=a \cdot f(x)$ scales the RANGE by a; domain is unchanged.

Horizontal dilation $g(x)$ =f(bx) scales the DOMAIN by $1/b$ ; range is unchanged.

Worked example

The graph of $f$ has domain $[-6, 9]$ and range $[-4, 8]$ . Let $g(x) = f(3x)$ . Find the domain of $g$ .

01
Horizontal compression by factor $b=3$ scales the domain by $1/3$ : divide both endpoints by 3.
02
Domain of $g$ : [−6/3, $9/3$ ] = $[-2, 3]$ . (Range stays $[-4, 8]$ , unaffected by a horizontal change.)

Practice Problems

Worked example

Problem 1. $f(x) = 4x - 1$ . Find $g(x) = 0.5f(x)$ .

01
$g(x) = 0.5(4x-1) = 2x - 0.5$ .

Worked example

Problem 2. $f(x) = x^{2} - 4$ . Find $g(x) = f(2x)$ , and describe the transformation.

01
$g(x) = (2x)^{2} - 4 = 4x^{2} - 4$ .
02
Since $b=2$ (|b|>1), this is a horizontal COMPRESSION — every x-value is squeezed toward the y-axis by a factor of 2.

Common slips

The single most common error: assuming f(bx) with $b>1$ stretches the graph. It actually compresses it — bigger $b$ squeezes the graph inward.

When $b$ is a fraction (like 0.5), the horizontal stretch factor is its reciprocal (here, stretched by 2, not by 0.5).

A negative $a$ or $b$ adds a reflection on top of the stretch/compression — handle the sign and the magnitude as two separate effects (see Section 4).
Don't confuse $g(x)=f(-x)$ (reflects over the y-axis, negative inside) with $g(x)=-f(x)$ (reflects over the x-axis, negative outside) — they do very different things.

When substituting $(-x)$ , apply the negative sign to every occurrence of $x$ , including inside exponents — $(-x)^{2} = x^{2}$ , but $(-x)^{3} = -x^{3}$ .

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3 cards · Transformations

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Recap card

6 lines to re-read the night before.

01
$g(x) = a \cdot f(x)$ : vertical dilation, matches intuition (bigger |a| → taller).
02
$g(x)$ = f(bx): horizontal dilation, opposite intuition (bigger |b| → narrower/compressed).
03
A horizontal stretch factor is the reciprocal of $b$ — don't confuse $b$ itself with the stretch amount.
04
Vertical dilation scales the range; horizontal dilation scales the domain (by $1/b$ ).
05
$g(x) = f(-x)$ : reflects over the y-axis (negative inside), distinct from $g(x) = -f(x)$ reflecting over the x-axis (negative outside).
06
AP notation f(x/k): a horizontal stretch by factor $k$ , the same as f(bx) with $b=1/k$ .

Practice 24 problems Next: Chapter 1.13 Function Model Selection and Assumption Articulation

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Dilations of Functions

Multiplicative Transformations: The Big Picture

Vertical Dilation: g(x) = a·f(x)

Horizontal Dilation: g(x) = f(bx)

Reflection Over the y-Axis: g(x) = f(−x)

Worked Example

Describing a Full Sequence of Transformations

Step-by-Step: Dilating Algebraically

Vertical dilation: g(x) = 3f(x)

Horizontal dilation: g(x) = f(3x)

Step-by-Step: Dilating Numerically

Domain and Range Under Dilation

Practice Problems

Recap card

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