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Transformations of Functions

Introduction

A function connects an input with an output, and its graph shows that relationship visually. In this aiMOOC, you will learn how to change a familiar graph in controlled ways without rebuilding it point by point. These changes are called transformations of functions. You will work with translations, reflections, stretches, and compressions, then combine them to analyze and create graphs.

This course is designed for Grades 9–10. You should already be comfortable with the coordinate plane, function notation such as $f (x)$ , and basic parent functions such as linear, quadratic, and absolute-value functions.

The image above shows several transformations of the absolute-value parent function. As you study, keep asking two questions: What changed in the equation? and What changed in the graph?

Learning Goals

By the end of this course, you should be able to:

Functions: Explain how a parent function provides a starting graph for a family of related functions.
Translations: Predict vertical and horizontal shifts from function notation.
Reflections: Identify reflections across the x-axis and y-axis.
Scaling: Distinguish vertical stretches, vertical compressions, horizontal stretches, and horizontal compressions.
Quadratic functions: Read transformations directly from vertex form.
Graphs of functions: Map key points from a parent graph to a transformed graph.
Modeling: Use transformations to fit and interpret simple real-world situations.

Parent Functions and Graph Families

A parent function is a simple function that represents the basic shape of a family of related graphs. Transformations change the position, orientation, or scale of that shape.

Common parent functions for Grades 9–10 include:

Parent function	Equation	Key feature
Linear	$f (x) = x$	A straight line through the origin
Quadratic	$f (x) = x^{2}$	A parabola with vertex at the origin
Absolute value	$f (x) = \| x \|$	A V-shaped graph with vertex at the origin
Cubic	$f (x) = x^{3}$	An S-shaped graph through the origin
Square root	$f (x) = \sqrt{x}$	Begins at the origin and extends to the right

A quadratic parent graph gives a particularly clear example because you can see shifts, reflections, and vertical scaling directly in vertex form.

The absolute-value graph is also useful because its vertex and two straight arms make changes in position and steepness easy to spot.

Vertical Translations

A vertical translation moves every point on a graph up or down by the same amount.

If $g (x) = f (x) + k$ ,

then:

If Fehler beim Parsen (Syntaxfehler): {\displaystyle k>0} , the graph moves up by $k$ units.
If Fehler beim Parsen (Syntaxfehler): {\displaystyle k<0} , the graph moves down by $| k |$ units.

For example, if $f (x) = x^{2}$ , then $g (x) = x^{2} + 3$ is the same parabola shifted up by three units. The vertex moves from $(0, 0)$ to $(0, 3)$ .

A vertical translation changes every output value by the same amount. The domain usually stays the same, while the range shifts up or down.

Horizontal Translations

A horizontal translation moves every point left or right.

If $g (x) = f (x - h)$ ,

then:

If Fehler beim Parsen (Syntaxfehler): {\displaystyle h>0} , the graph moves right by $h$ units.
If the input is written as $x + h$ , the graph moves left by $h$ units.

The sign can feel backward because the change happens inside the function input. A useful point rule is:

$(x, y) \to (x + h, y)$ for $g (x) = f (x - h)$ .

For example, $g (x) = (x - 4)^{2}$ is the graph of $y = x^{2}$ shifted right by four units.

A translation preserves the graph's shape. Only its location changes.

Reflections

A reflection flips a graph across a line.

For a reflection across the x-axis: $g (x) = - f (x)$ .

Every y-value changes sign, so a point $(x, y)$ becomes $(x, - y)$ .

For a reflection across the y-axis: $g (x) = f (- x)$ .

Every x-value changes sign, so a point $(x, y)$ becomes $(- x, y)$ .

For example, the graph of $y = - x^{2}$ is the reflection of $y = x^{2}$ across the x-axis. It opens downward instead of upward.

Vertical Stretches and Compressions

If $g (x) = a f (x)$ ,

then every output value is multiplied by $a$ .

When Fehler beim Parsen (Syntaxfehler): {\displaystyle |a|>1} , the graph is stretched vertically. Points move farther from the x-axis.

When Fehler beim Parsen (Syntaxfehler): {\displaystyle 0<|a|<1} , the graph is compressed vertically. Points move closer to the x-axis.

When Fehler beim Parsen (Syntaxfehler): {\displaystyle a<0} , the graph is also reflected across the x-axis.

For the quadratic parent $f (x) = x^{2}$ :

$g (x) = 3 x^{2}$ is narrower because of a vertical stretch by a factor of three.
$g (x) = \frac{1}{2} x^{2}$ is wider because of a vertical compression by a factor of one half.
$g (x) = - 2 x^{2}$ is vertically stretched by a factor of two and reflected across the x-axis.

The family of parabolas above illustrates how multiplying the output changes the width of a quadratic graph.

Horizontal Stretches and Compressions

If $g (x) = f (b x)$ ,

then x-coordinates are divided by $b$ .

For positive $b$ :

If Fehler beim Parsen (Syntaxfehler): {\displaystyle b>1} , the graph is compressed horizontally by a factor of $\frac{1}{b}$ .
If Fehler beim Parsen (Syntaxfehler): {\displaystyle 0<b<1} , the graph is stretched horizontally by a factor of $\frac{1}{b}$ .

If Fehler beim Parsen (Syntaxfehler): {\displaystyle b<0} , there is also a reflection across the y-axis.

The inside factor works reciprocally. For example, $g (x) = f (2 x)$ is a horizontal compression by a factor of one half, not a stretch by a factor of two.

A General Transformation Rule

Many transformations can be organized with the form

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

Each parameter has a job:

Parameter	Main effect	Graph interpretation
$a$	Multiplies outputs	Vertical scale by $\| a \|$ ; reflect across the x-axis if negative
$b$	Multiplies inputs	Horizontal scale by $\frac{1}{\| b \|}$ ; reflect across the y-axis if negative
$h$	Changes input location	Shift right by $h$ when written as $x - h$
$k$	Changes output location	Shift up by $k$

A reliable point-mapping rule is especially useful. If $(u, v)$ lies on $y = f (x)$ , then the corresponding point on $y = a f (b (x - h)) + k$ is

$(h + \frac{u}{b}, k + a v)$ ,

provided $b \neq 0$ .

This rule helps you avoid memorizing a complicated order of operations.

Quadratic Functions in Vertex Form

A quadratic function in vertex form is

$y = a (x - h)^{2} + k$ .

It is a transformation of the parent function $y = x^{2}$ .

You can read several features immediately:

The vertex is $(h, k)$ .
The axis of symmetry is $x = h$ .
If Fehler beim Parsen (Syntaxfehler): {\displaystyle a>0} , the parabola opens upward.
If Fehler beim Parsen (Syntaxfehler): {\displaystyle a<0} , the parabola opens downward.
If Fehler beim Parsen (Syntaxfehler): {\displaystyle |a|>1} , the parabola is narrower than $y = x^{2}$ .
If Fehler beim Parsen (Syntaxfehler): {\displaystyle 0<|a|<1} , the parabola is wider than $y = x^{2}$ .

Example: For $y = - 2 (x - 3)^{2} + 1$ , start with $y = x^{2}$ . Shift right three units, stretch vertically by a factor of two, reflect across the x-axis, and shift up one unit. The vertex is $(3, 1)$ .

Worked Examples

Example 1: Translation and vertical stretch

Start with $f (x) = x^{2}$ and define $g (x) = 2 (x - 3)^{2} - 1$ . The graph shifts right three units, stretches vertically by a factor of two, and shifts down one unit. The vertex becomes $(3, - 1)$ .

Example 2: Absolute value with reflection

Start with $f (x) = | x |$ and define $g (x) = - | x + 2 | + 4$ . The graph shifts left two units, reflects across the x-axis, and shifts up four units. Its vertex is $(- 2, 4)$ .

Example 3: Factoring the input first

Suppose $g (x) = f (2 x - 4) + 1$ . First rewrite the input as $2 (x - 2)$ . Now you can see a horizontal compression by a factor of one half, a shift right two units, and a shift up one unit. Reading the unfactored expression too quickly can lead to the incorrect claim that the graph shifts right four units.

For a longer guided review that connects shifts, reflections, and scaling, use this lesson:

Transforming Key Features

Transformations affect more than the shape of a graph. They also affect important features.

Vertices and turning points move according to the same point-mapping rules as all other points.

Domain changes under horizontal transformations. If the parent function begins at a certain x-value, a horizontal shift or scale changes that starting x-value.

Range changes under vertical transformations. A vertical shift changes all y-values, while a negative vertical factor can reverse maximum and minimum behavior.

Intercepts may change in different ways. A vertical stretch keeps x-intercepts fixed when the scale factor is nonzero, but a vertical shift can create or remove x-intercepts. Horizontal transformations move x-intercepts according to the input rule.

Symmetry can be preserved or moved. A quadratic parent function is symmetric about the y-axis, while a translated quadratic is symmetric about the vertical line through its new vertex.

Applications and Modeling

Function transformations are useful when a basic model has the right shape but the wrong position or scale.

A quadratic parent function can be shifted and stretched to model the height of an arch or the path of an object in a simplified setting. An absolute-value function can model distance from a central location. A square-root function can be shifted to represent a process that begins later than time zero. In each case, transformations let you adapt a familiar function instead of inventing a new model from scratch.

When you model a situation, interpret the parameters in context. A vertical shift may represent a baseline height, a horizontal shift may represent a delayed starting time, and a scale factor may represent a change in size or rate.

Common Mistakes and How to Avoid Them

Mistake 1: Reversing horizontal shift signs. Remember that $f (x - h)$ moves right when $h$ is positive.

Mistake 2: Treating inside and outside factors the same way. Outside factors multiply y-values directly. Inside factors change x-values reciprocally.

Mistake 3: Forgetting to factor the input. In an expression such as $f (2 x - 4)$ , rewrite it as $f (2 (x - 2))$ before describing the shift.

Mistake 4: Confusing reflection axes. A negative sign outside the function reflects across the x-axis. A negative sign on the input reflects across the y-axis.

Mistake 5: Transforming only one special point. Check several points or use the point-mapping rule so that the entire graph stays consistent.

Interactive Tasks

Quiz: Test Your Knowledge

If g(x)=f(x)+5, what happens to the graph of f? (It shifts up five units) (!It shifts down five units) (!It shifts right five units) (!It reflects across the x-axis)

If g(x)=f(x-3), what happens to the graph of f? (It shifts right three units) (!It shifts left three units) (!It shifts up three units) (!It is compressed vertically)

What transformation is produced by g(x)=-f(x)? (A reflection across the x-axis) (!A reflection across the y-axis) (!A shift to the left) (!A horizontal stretch)

What transformation is produced by g(x)=f(-x)? (A reflection across the y-axis) (!A reflection across the x-axis) (!A shift upward) (!A vertical compression)

What does g(x)=3f(x) do to the graph of f? (It stretches the graph vertically by a factor of three) (!It shifts the graph up three units) (!It stretches the graph horizontally by a factor of three) (!It shifts the graph right three units)

What does g(x)=0.5f(x) do to the graph of f? (It compresses the graph vertically) (!It reflects the graph across the y-axis) (!It shifts the graph down) (!It compresses the graph horizontally)

What does g(x)=f(2x) do to the graph of f? (It compresses the graph horizontally by a factor of one half) (!It stretches the graph horizontally by a factor of two) (!It shifts the graph right two units) (!It stretches the graph vertically by a factor of two)

What is the vertex of y=-2(x-4)^2+1? (The vertex has coordinates four comma one) (!The vertex has coordinates negative four comma one) (!The vertex has coordinates four comma negative one) (!The vertex has coordinates two comma one)

In g(x)=a f(b(x-h))+k, which parameter controls the vertical shift? (The parameter k) (!The parameter a) (!The parameter b) (!The parameter h)

Why is factoring the input useful in f(2x-4)? (It reveals the horizontal scale and shift clearly) (!It changes a vertical shift into a reflection) (!It removes the need to graph the function) (!It guarantees that the function is quadratic)

Memory Game

Parent function	A simple starting graph for a family of related functions
Vertical translation	A movement of the graph up or down
Horizontal translation	A movement of the graph left or right
X-axis reflection	A flip that changes the sign of every output
Y-axis reflection	A flip that changes the sign of every input coordinate
Vertical stretch	A scaling that moves points farther from the x-axis
Vertical compression	A scaling that moves points closer to the x-axis
Horizontal compression	A scaling that moves points closer to the y-axis

Drag and Drop

Match the correct terms.	Topic
Shift upward	Add a positive constant outside the function
Shift right	Subtract a positive constant from the input
Reflect across x-axis	Multiply the entire output by a negative sign
Vertical stretch	Multiply outputs by a factor greater than one
Horizontal compression	Multiply the input by a factor greater than one

...

Crossword Puzzle

Translation	What transformation slides a graph without changing its shape?
Reflection	What transformation flips a graph across an axis?
Compression	What scaling makes a graph closer to an axis?
Stretching	What scaling makes distances from an axis larger?
Parabola	What is the graph shape of the quadratic parent function?
Vertex	What is the turning point of a parabola called?

LearningApps

Cloze Text

Open-Ended Tasks

Easy

Graph Sketch Gallery: Draw a parent function and three transformed versions of it, label each equation, and write one sentence explaining every change.
Transformation Vocabulary Cards: Create a set of illustrated cards for translation, reflection, stretch, and compression, using one original example on each card.
Digital Slider Experiment: Use a graphing tool to vary one parameter in a transformed function, record at least five observations, and explain the pattern you see.
Mini Explainer Video: Record a short video in which you teach the difference between an inside change and an outside change in function notation.

Standard

Quadratic Design Challenge: Design three parabolas from $y = x^{2}$ using different values of a, h, and k, then explain how each parameter affects the graph.
Peer Transformation Interview: Interview a classmate about one transformation rule, ask them to explain an example, and write a short reflection comparing their explanation with your own.
Graph Error Detective: Create a deliberately incorrect transformation solution, exchange it with a partner, and write a correction that explains exactly where the reasoning fails.
Motion Model Experiment: Collect or simulate a small set of position data, choose a familiar parent-function shape, and test how shifts or scale factors improve the fit.

Advanced

Transformation Composition Investigation: Compare two different orders of transformations on the same parent function and determine which operations commute and which do not.
Real-World Modeling Project: Find or create data that resembles a quadratic, absolute-value, or square-root graph, fit a transformed parent function, and interpret every parameter in context.
Point Mapping Proof: Use the general form $g (x) = a f (b (x - h)) + k$ to derive the point-mapping rule and verify it with at least three parent-function points.
Mathematical Photo Trail: Visit your school, neighborhood, museum, sports area, or another safe public place, photograph shapes that can be modeled by transformed functions, and build a labeled digital gallery with proposed equations.

Learning Assessment

Transformation Reasoning: Given a graph and a parent function, explain which sequence of transformations could produce the graph and justify each step from visible features.
Equation Construction: Build an equation for a transformed function that satisfies a stated vertex, orientation, and scale, then explain why your equation meets all conditions.
Error Analysis: Analyze a solution that claims $f (x + 4)$ shifts a graph right four units, identify the misconception, and correct the reasoning with a point example.
Feature Transfer: Predict how the domain, range, intercepts, and symmetry of a parent function change under a specified transformation without graphing every point.
Model Comparison: Compare two transformed functions that could fit the same real-world situation and argue which model is more reasonable based on the meaning of its parameters.
General Rule Application: Use the point-mapping rule for $g (x) = a f (b (x - h)) + k$ to transform several key points and check the result against the transformed equation.

Evidence of Learning

Knowledge: You can explain parent functions, translations, reflections, vertical and horizontal scaling, vertex form, and the role of the parameters a, b, h, and k.
Skills: You can predict transformations from equations, write equations from graphs, map key points, identify common errors, and explain how domain and range respond to transformations.
Products: Useful evidence includes accurate graph sketches, digital graphs, annotated models, short explanatory videos, investigation notes, and completed projects.
Transfer achievements: You can recognize transformed parent-function shapes in new contexts, select an appropriate model, interpret parameters meaningfully, and defend your reasoning with equations and graph features.

OERs on the Topic

For broader background on graphs of functions, use the English Wikipedia article below. Connect its ideas about ordered pairs, coordinates, domain, range, and graphs to the transformations in this course.