Topic 1.12A · CED: Transformations of Functions

Translations of Functions

6 MIN READ10 IDEAS23 PROBLEMS3 flashcards

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01
$g(x) = f(x) + k$ : $(x, y)$ on $f$ becomes $(x, y+k)$ on $g$ — $x$ stays the same, $y$ shifts by $k$ .
02
$g(x) = f(x+h)$ : $(x, y)$ on $f$ becomes $(x-h, y)$ on $g$ — the graph shifts opposite the sign of $h$ .
03
$g(x) = -f(x)$ : $(x, y)$ on $f$ becomes $(x, -y)$ on $g$ .

Practice 23 problems Next: Chapter 1.12B Dilations of Functions

Additive Transformations: The Big Picture

CONCEPT

Two Kinds of Shifts

$g(x) = f(x) + k$ → VERTICAL shift (moves up/down) — matches intuition.

$g(x) = f(x + h)$ → HORIZONTAL shift (moves left/right) — the direction is the OPPOSITE of what the sign suggests.

The rest of this note is built around ONE generic function $f(x)$ , so you can watch exactly how each transformation moves every point on the graph.

Vertical Shifts: g(x) = f(x) + k

Adding a number OUTSIDE the function shifts every point straight up or down by that amount. The x-coordinates never change — only the y-coordinates.

Every point on $f$ moves straight up by 4 — the x-coordinate stays the same, only $y$ changes.

Subtracting moves every point straight down instead.

KEY RULE

$g(x) = f(x) + k$ : $(x, y)$ on $f$ becomes $(x, y+k)$ on $g$ — $x$ stays the same, $y$ shifts by $k$ .

Horizontal Shifts: g(x) = f(x + h)

This is the one that trips everybody up: $f(x + 4)$ shifts LEFT, not right — even though the sign inside looks like it should add. Here's why: to get the SAME output $f$ used to give at some input, $g$ now needs an input that's 4 SMALLER (since $g(x)=f(x+4)$ means you're evaluating $f$ at $x+4$ , which reaches any given target 4 units earlier).

$f(x+4)$ — every point moves LEFT by 4. Notice the y-coordinates never change.

$f(x-4)$ — every point moves RIGHT by 4.

KEY RULE

$g(x) = f(x+h)$ : $(x, y)$ on $f$ becomes $(x-h, y)$ on $g$ — the graph shifts OPPOSITE the sign of $h$ .

Quick check

The point $(5, -2)$ is on the graph of $f$ . Where does it land on $g(x) = f(x + 1) - 3$ ?

Vertical Reflection: g(x) = −f(x)

Multiplying the WHOLE function by −1 flips every point over the x-axis — like a mirror. x-coordinates stay put; y-coordinates flip sign.

Each point's height flips sign — a point above the x-axis lands the same distance below, and vice versa.

KEY RULE

$g(x) = -f(x)$ : $(x, y)$ on $f$ becomes $(x, -y)$ on $g$ .

Combining Transformations

Transformations can stack. For $g(x) = -f(x-2) + 5$ , here's the key idea: horizontal changes and vertical changes are completely INDEPENDENT of each other — neither one has to "wait" for the other.

CONCEPT

How to Read the Stack

Horizontal side: $x-2$ shifts every point RIGHT 2 — that's the only horizontal change here.

Vertical side: −(...) + 5 has TWO vertical operations. Follow order of operations: multiply/flip the sign FIRST, then add — so reflect over the x-axis first, then shift up 5.

Point $(-2,4)$ on $f$ becomes $(0,1)$ on $g$ ; point $(2,5)$ on $f$ becomes $(4,0)$ on $g$ — shift right 2 (horizontal, independent), reflect then shift up 5 (vertical, in that order).

KEY RULE

Horizontal and vertical transformations are independent — apply them in either order. WITHIN the vertical side, multiply/reflect FIRST, then add — matching standard order of operations.

COMMON MISTAKE

Don't apply the horizontal shift with the same intuition as the vertical shift — $f(x+h)$ moves opposite the sign, $f(x)$ +k moves with the sign.

On the vertical side specifically, don't add before you multiply/reflect — $g(x) = -f(x-2)+5$ means $(-1) \cdot f(x-2)$ , THEN +5, not the other way around.

Quick check

$(-1, 4)$ is on $f$ . Where is it on $g(x) = -f(x - 2) + 1$ ?

Step-by-Step: Transforming Algebraically

Worked example

Given $f(x) = x^{2} - 5x + 4$ , find $g(x) = f(x) + 6$ , and separately $g(x) = f(x + 6)$ .

Vertical shift: g(x) = f(x) + 6

01
Just add 6 to the entire function — no need to touch the x's:

g(x) = f(x) + 6 = x^{2} - 5x + 10

Horizontal shift: g(x) = f(x + 6)

01
Replace every $x$ in $f(x)$ with $(x+6)$ :

g(x) = f(x + 6) = (x + 6)^{2} - 5(x + 6) + 4

02
Expand and simplify:

g(x) = x^{2} + 7x + 10

Step-by-Step: Transforming Numerically

Worked example

Given the table of values for $f$ below, let $g(x) = f(x - 3) + 2$ . Find $g(5)$ .

x	f(x)
0	6
1	9
2	3
3	1

01
$g(5) = f(5-3) + 2 = f(2) + 2$ — first find $x$ such that shifting lands on a value in the table.
02
From the table, $f(2) = 3$ .
03
$g(5) = 3 + 2 = 5$ .

Domain and Range Under Transformation

CONCEPT

The Rule

Vertical shifts and reflections change the RANGE (the outputs) — the domain stays the same.

Horizontal shifts change the DOMAIN (the inputs) — the range stays the same.

Reflection over the x-axis flips the range's inequality direction (and its sign).

Worked example

The graph of $f$ has domain $[-6, 2]$ and range $(-3, 7)$ . Let $g(x) = -f(x+5) + 1$ . Find the domain and range of $g$ .

01
Horizontal shift by −5 (i.e., LEFT 5) affects the domain: subtract 5 from both endpoints: [−6−5, 2−5] = $[-11, -3]$ .
02
Reflection flips the range's sign: $(-3,7)$ becomes $(-7,3)$ .
03
Vertical shift by +1 adds 1 to the (already flipped) range: $(-7+1, 3+1) = (-6, 4)$ .
04
Domain of $g$ : $[-11, -3]$ . Range of $g$ : $(-6, 4)$ .

Practice Problems

Worked example

Problem 1. $f(x) = 3x - 2$ . Find $g(x) = f(x) + 7$ .

01
$g(x) = (3x-2) + 7 = 3x + 5$ .

Worked example

Problem 2. $f(x) = 2x^{2} + x$ . Find $g(x) = f(x - 3)$ .

01
$g(x) = 2(x-3)^{2} + (x-3) = 2(x^{2}-6x+9) + x - 3 = 2x^{2}-12x+18+x-3 = 2x^{2}-11x+15$ .

Common slips

Don't apply the horizontal shift with the same intuition as the vertical shift — $f(x+h)$ moves opposite the sign, $f(x)$ +k moves with the sign.

On the vertical side specifically, don't add before you multiply/reflect — $g(x) = -f(x-2)+5$ means $(-1) \cdot f(x-2)$ , then +5, not the other way around.

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3 cards · Transformations

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Recap card

5 lines to re-read the night before.

01
$g(x) = f(x) + k$ : vertical shift, matches intuition ( $k>0$ → up).
02
$g(x) = f(x + h)$ : horizontal shift, opposite the sign ( $h>0$ → left).
03
$g(x) = -f(x)$ : reflects over the x-axis — y-values flip sign, x-values unchanged.
04
Stacked transformations apply in the order the function is built: innermost operation on $x$ first.
05
Horizontal changes affect domain; vertical changes (including reflection) affect range.

Practice 23 problems Next: Chapter 1.12B Dilations of Functions

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Translations of Functions

Additive Transformations: The Big Picture

Vertical Shifts: g(x) = f(x) + k

Horizontal Shifts: g(x) = f(x + h)

Vertical Reflection: g(x) = −f(x)

Combining Transformations

Step-by-Step: Transforming Algebraically

Vertical shift: g(x) = f(x) + 6

Horizontal shift: g(x) = f(x + 6)

Step-by-Step: Transforming Numerically

Domain and Range Under Transformation

Practice Problems

Recap card

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