Transformations of Functions

You have been transforming things your entire life without calling it that. When you adjust the brightness on your phone, you are stretching or compressing the light values. When you move an app icon from one spot to another on your screen, you are translating it. When you flip a photo horizontally before posting it, you are reflecting it. These are all transformations, and in mathematics, we do the exact same things to functions. The beautiful part is that once you understand how transformations work, you can graph almost any function quickly by starting with a simple “parent” function and then shifting, stretching, or flipping it into place. No need to plot point after tedious point.

Core Concepts

Parent Functions: Your Starting Point

A parent function is the simplest, most basic version of a function family. Think of it as the original template before any modifications. Just as you might start with a basic photo and then apply filters and adjustments, you start with a parent function and then apply transformations.

Here are the parent functions you will work with most often:

Family	Parent Function	Key Features
Linear	$f(x) = x$	Straight line through the origin, slope of 1
Quadratic	$f(x) = x^2$	U-shaped parabola, vertex at the origin
Absolute Value	$f(x) = |x|$	V-shape, vertex at the origin
Square Root	$f(x) = \sqrt{x}$	Starts at origin, curves upward to the right
Cubic	$f(x) = x^3$	S-shaped curve through the origin

Why do parent functions matter? Because once you memorize these basic shapes, you can graph any transformed version of them without plotting individual points. You simply apply the transformations to shift, stretch, or flip the parent function into its new position.

Vertical Translations: Moving Up and Down

The simplest transformation is moving a function straight up or down. This is called a vertical translation (or vertical shift).

If you add a number to your function, the graph moves up. If you subtract a number, the graph moves down. The pattern is:

$$f(x) + k \text{ shifts the graph UP by } k \text{ units}$$ $$f(x) - k \text{ shifts the graph DOWN by } k \text{ units}$$

Think about it this way: if $f(x) = x^2$ gives you the output 4 when you input 2, then $f(x) + 3 = x^2 + 3$ gives you $4 + 3 = 7$ for the same input. Every single output gets 3 added to it, which moves every point up by 3.

This makes intuitive sense. Adding to the output directly affects the $y$-values, and $y$-values control vertical position.

Horizontal Translations: Moving Left and Right

Moving a function left or right is called a horizontal translation (or horizontal shift). This one trips people up because it works opposite to what you might expect:

$$f(x - h) \text{ shifts the graph RIGHT by } h \text{ units}$$ $$f(x + h) \text{ shifts the graph LEFT by } h \text{ units}$$

Wait, subtracting shifts right? That seems backward. Here is why it works this way:

Consider $(x - 3)^2$. To get the same output that the parent function $x^2$ gives at $x = 0$ (which is 0), you now need to input $x = 3$ because $(3 - 3)^2 = 0^2 = 0$. The output that used to happen at $x = 0$ now happens at $x = 3$. The graph moved right by 3.

The key insight: horizontal transformations work on the input before the function processes it. When you write $f(x - 3)$, you are saying “first subtract 3 from $x$, then apply the function.” This delays when things happen, pushing the graph to the right.

A helpful memory trick: the horizontal shift is the value of $x$ that makes the inside of the parentheses equal zero. For $(x - 3)^2$, solving $x - 3 = 0$ gives $x = 3$, so the shift is 3 units right.

Vertical Stretches and Compressions

Multiplying the entire function by a number changes how tall or short the graph appears. This is called a vertical stretch or vertical compression.

$$a \cdot f(x) \text{ where } |a| > 1 \text{ stretches the graph vertically (makes it taller)}$$ $$a \cdot f(x) \text{ where } 0 < |a| < 1 \text{ compresses the graph vertically (makes it shorter)}$$

For example, $2x^2$ is twice as tall as $x^2$ at every point. When the parent function outputs 4, the stretched version outputs 8. When $x = 1$, the parent gives $1^2 = 1$, but $2x^2$ gives $2(1)^2 = 2$. Every $y$-value is doubled.

Similarly, $\frac{1}{2}x^2$ is half as tall as $x^2$. All the $y$-values are halved.

Horizontal Stretches and Compressions

You can also stretch or compress a function horizontally by multiplying the input:

$$f(bx) \text{ where } |b| > 1 \text{ compresses the graph horizontally (makes it narrower)}$$ $$f(bx) \text{ where } 0 < |b| < 1 \text{ stretches the graph horizontally (makes it wider)}$$

Again, this works opposite to what you might expect. A bigger multiplier on $x$ makes the graph narrower, not wider.

Here is why: consider $(2x)^2$ compared to $x^2$. To get the output that the parent function gives at $x = 2$ (which is 4), you only need to input $x = 1$ because $(2 \cdot 1)^2 = 2^2 = 4$. Everything happens sooner, compressing the graph horizontally.

The horizontal stretch/compression factor is actually $\frac{1}{b}$. So $f(2x)$ compresses by a factor of $\frac{1}{2}$ (half as wide), and $f(\frac{1}{2}x)$ stretches by a factor of 2 (twice as wide).

Reflections: Flipping the Graph

Reflections flip the graph over an axis. There are two types:

Reflection over the x-axis: $-f(x)$

This negates all the outputs, flipping the graph upside down. If a point was at $(2, 4)$, it moves to $(2, -4)$. All positive $y$-values become negative and vice versa.

For example, $-x^2$ is the parabola $x^2$ flipped upside down. Instead of opening upward, it opens downward.

Reflection over the y-axis: $f(-x)$

This negates all the inputs, flipping the graph left-to-right. If a point was at $(3, 5)$, it moves to $(-3, 5)$.

For example, $\sqrt{-x}$ is the square root function flipped horizontally. Instead of extending to the right from the origin, it extends to the left.

Note: Some functions look the same when reflected over the y-axis. These are called even functions (like $x^2$, since $(-x)^2 = x^2$). Some functions flip upside down when reflected over the y-axis. These are called odd functions (like $x^3$, since $(-x)^3 = -x^3$).

Combining Multiple Transformations

Real problems often involve multiple transformations at once. The general form for combining transformations is:

$$g(x) = a \cdot f(b(x - h)) + k$$

where:

• $a$ controls vertical stretch/compression and reflection over x-axis
• $b$ controls horizontal stretch/compression and reflection over y-axis
• $h$ controls horizontal shift (right if positive, left if negative)
• $k$ controls vertical shift (up if positive, down if negative)

Order of Operations for Graphing:

When graphing, apply transformations in this order:

1. Start with the parent function
2. Apply horizontal stretch/compression and horizontal reflection (the $b$)
3. Apply vertical stretch/compression and vertical reflection (the $a$)
4. Apply horizontal shift (the $h$)
5. Apply vertical shift (the $k$)

A helpful way to remember: work from the “inside out” for horizontal changes, then handle vertical changes, and save shifts for last.

Writing Equations from Transformed Graphs

When you see a graph and need to write its equation:

1. Identify the parent function by looking at the basic shape (parabola, V-shape, curve, etc.)
2. Find key points like the vertex or starting point
3. Compare to the parent to determine shifts, stretches, and reflections
4. Build the equation piece by piece

For a parabola that opens downward with vertex at $(2, 5)$:

• Parent function: $f(x) = x^2$
• Opens downward: needs $-$ sign (reflection)
• Vertex at $(2, 5)$: shifted right 2 (gives $x - 2$) and up 5 (gives $+ 5$)
• Basic equation: $g(x) = -(x - 2)^2 + 5$

To find any vertical stretch, use another point on the graph and solve for $a$.

Notation and Terminology

Term	Meaning	Example
Parent function	Simplest form of a function family	$f(x) = x^2$ for quadratics
$f(x) + k$	Vertical shift up by $k$	$x^2 + 3$ shifts up 3
$f(x) - k$	Vertical shift down by $k$	$x^2 - 3$ shifts down 3
$f(x - h)$	Horizontal shift right by $h$	$(x-2)^2$ shifts right 2
$f(x + h)$	Horizontal shift left by $h$	$(x+2)^2$ shifts left 2
$af(x)$ where $|a| > 1$	Vertical stretch by factor of $|a|$	$3x^2$ stretches vertically by 3
$af(x)$ where $0 < |a| < 1$	Vertical compression by factor of $|a|$	$\frac{1}{2}x^2$ compresses vertically
$f(bx)$ where $|b| > 1$	Horizontal compression by factor of $\frac{1}{|b|}$	$(2x)^2$ compresses horizontally
$f(bx)$ where $0 < |b| < 1$	Horizontal stretch by factor of $\frac{1}{|b|}$	$(\frac{1}{2}x)^2$ stretches horizontally
$-f(x)$	Reflection over x-axis	$-x^2$ flips upside down
$f(-x)$	Reflection over y-axis	$(-x)^2$ flips left-to-right

Examples

Key Properties and Rules

Transformation Quick Reference

Transformation	What It Does	Notation
Vertical shift up $k$	Moves graph up	$f(x) + k$
Vertical shift down $k$	Moves graph down	$f(x) - k$
Horizontal shift right $h$	Moves graph right	$f(x - h)$
Horizontal shift left $h$	Moves graph left	$f(x + h)$
Vertical stretch	Makes graph taller	$af(x)$ where $|a| > 1$
Vertical compression	Makes graph shorter	$af(x)$ where $0 < |a| < 1$
Horizontal compression	Makes graph narrower	$f(bx)$ where $|b| > 1$
Horizontal stretch	Makes graph wider	$f(bx)$ where $0 < |b| < 1$
Reflection over x-axis	Flips graph upside down	$-f(x)$
Reflection over y-axis	Flips graph left-to-right	$f(-x)$

The “Inside vs. Outside” Rule

Transformations that happen inside the function (affecting $x$) work horizontally and often seem “backward”:

• Adding inside shifts left
• Subtracting inside shifts right
• Multiplying $x$ by a number greater than 1 compresses horizontally

Transformations that happen outside the function (affecting the output) work vertically and are more intuitive:

• Adding outside shifts up
• Subtracting outside shifts down
• Multiplying outside by a number greater than 1 stretches vertically

Order of Operations for Graphing

When graphing a transformed function:

1. Start with the parent function
2. Apply horizontal transformations (stretch/compression, then shift)
3. Apply vertical transformations (stretch/compression, then shift)
4. Reflections can be applied with their respective stretch/compression

Domain and Range Changes

Transformations affect domain and range predictably:

Transformation	Effect on Domain	Effect on Range
Vertical shift by $k$	No change	Shifts by $k$
Horizontal shift by $h$	Shifts by $h$	No change
Vertical stretch/compression	No change	Stretched/compressed
Horizontal stretch/compression	Stretched/compressed	No change
Reflection over x-axis	No change	Reflected
Reflection over y-axis	Reflected	No change

Real-World Applications

Adjusting Mathematical Models

Scientists and economists often start with a basic model and then adjust it using transformations. For example, a basic exponential growth model might be shifted up to account for an initial population, shifted right to account for when measurement began, or stretched vertically to account for a faster growth rate.

If a biologist discovers that a population follows the pattern $P(t) = 2^t$ but the population was already 500 when observation began at year 2010, the adjusted model might be $P(t) = 2^{t-2010} + 500$. The horizontal shift accounts for the starting year, and the vertical shift accounts for the initial population.

Image Manipulation in Computer Graphics

Every photo editing tool you use applies transformations. When you:

• Move an image: Translation (horizontal and/or vertical shift)
• Resize an image: Stretch or compression
• Flip an image: Reflection
• Rotate an image: A combination of transformations

Video game graphics work the same way. Characters and objects are defined by basic shapes that are then transformed (moved, scaled, rotated) thousands of times per second to create animation.

Sound Wave Modifications

Sound is represented mathematically as waves (often sine and cosine functions). When you adjust sound:

• Volume control: Vertical stretch or compression (larger amplitude = louder)
• Pitch shift: Horizontal stretch or compression (compressed waves = higher pitch)
• Phase shift: Horizontal translation (shifts when the sound wave peaks)
• Inversion: Reflection over the x-axis (flips the wave, useful in noise cancellation)

Noise-canceling headphones work by detecting incoming sound waves and generating the reflection (negative) of those waves. When the original wave and its reflection combine, they cancel out, reducing the noise you hear.

Economic Models

Economists use transformations to adapt basic models to different situations:

• A supply curve might shift right when new technology makes production easier
• A demand curve might shift up when consumer income increases
• Production functions might be scaled to account for differences in firm size

The mathematical form $Q = a \cdot f(bL - h) + k$ might represent output $Q$ as a function of labor $L$, with parameters adjusted for different industries or time periods.

Self-Test Problems

Summary

• Parent functions are the simplest forms of function families (linear, quadratic, absolute value, square root, cubic). Learn their shapes well since all transformed versions start here.

• Vertical translations ($f(x) + k$ or $f(x) - k$) move the graph up or down. Adding moves up; subtracting moves down.

• Horizontal translations ($f(x - h)$ or $f(x + h)$) move the graph left or right. Subtracting inside moves right; adding inside moves left. This seems backward because horizontal transformations work on the input.

• Vertical stretches and compressions ($a \cdot f(x)$) make the graph taller or shorter. Multiply by a number greater than 1 to stretch; multiply by a number between 0 and 1 to compress.

• Horizontal stretches and compressions ($f(bx)$) make the graph wider or narrower. The compression factor is $\frac{1}{b}$, which again seems backward.

• Reflections flip the graph. Negate the output ($-f(x)$) to reflect over the x-axis. Negate the input ($f(-x)$) to reflect over the y-axis.

• Combining transformations: The general form is $g(x) = a \cdot f(b(x - h)) + k$. Apply transformations systematically: horizontal changes first (inside the function), then vertical changes (outside the function).

• Writing equations from graphs: Identify the parent function, find key points like the vertex, determine shifts and stretches by comparing to the parent, and use additional points to find unknown parameters.

• Graphing efficiency: Instead of plotting many points, start with a known parent function and transform it. This is faster and gives you better insight into the function’s behavior.

• Transformations appear everywhere in the real world: adjusting scientific models, manipulating images and sound, and adapting economic functions to different contexts.