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{{T}}
[[Category:English]]
[[Category:Transformations of Functions]]

{{BR}}
= 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 <math>f(x)</math>, and basic parent functions such as linear, quadratic, and absolute-value functions.

[[File:GeometryTransformations.svg|500px|frameless|center]]

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?'''

{{#ev:youtube|https://www.youtube.com/watch?v=RttvubuBhAE|500|center}}

{{BR}}
== Learning Goals ==

By the end of this course, you should be able to:
# [[English:Function|Functions]]: Explain how a parent function provides a starting graph for a family of related functions.
# [[English:Translation|Translations]]: Predict vertical and horizontal shifts from function notation.
# [[English:Reflection|Reflections]]: Identify reflections across the x-axis and y-axis.
# [[English:Scaling|Scaling]]: Distinguish vertical stretches, vertical compressions, horizontal stretches, and horizontal compressions.
# [[English:Quadratic function|Quadratic functions]]: Read transformations directly from vertex form.
# [[English:Graph of a function|Graphs of functions]]: Map key points from a parent graph to a transformed graph.
# [[English:Mathematical modeling|Modeling]]: Use transformations to fit and interpret simple real-world situations.

{{BR}}
= 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:

{| class="wikitable"
! Parent function
! Equation
! Key feature
|-
| [[English:Linear function|Linear]]
| <math>f(x)=x</math>
| A straight line through the origin
|-
| [[English:Quadratic function|Quadratic]]
| <math>f(x)=x^2</math>
| A parabola with vertex at the origin
|-
| [[English:Absolute value|Absolute value]]
| <math>f(x)=|x|</math>
| A V-shaped graph with vertex at the origin
|-
| [[English:Cubic function|Cubic]]
| <math>f(x)=x^3</math>
| An S-shaped graph through the origin
|-
| [[English:Square root|Square root]]
| <math>f(x)=\sqrt{x}</math>
| Begins at the origin and extends to the right
|}

[[File:Parabola.svg|420px|frameless|center]]

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

[[File:Absolute Value.svg|500px|frameless|center]]

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

{{BR}}
= Vertical Translations =

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

If
<math>g(x)=f(x)+k</math>,

then:
# If <math>k>0</math>, the graph moves up by <math>k</math> units.
# If <math>k<0</math>, the graph moves down by <math>|k|</math> units.

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

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

{{BR}}
= Horizontal Translations =

A horizontal translation moves every point left or right.

If
<math>g(x)=f(x-h)</math>,

then:
# If <math>h>0</math>, the graph moves right by <math>h</math> units.
# If the input is written as <math>x+h</math>, the graph moves left by <math>h</math> units.

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

<math>(x,y)\rightarrow(x+h,y)</math> for <math>g(x)=f(x-h)</math>.

For example, <math>g(x)=(x-4)^2</math> is the graph of <math>y=x^2</math> shifted right by four units.

[[File:Translated graph of a function.png|500px|frameless|center]]

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

{{BR}}
= Reflections =

A reflection flips a graph across a line.

For a reflection across the '''x-axis''':
<math>g(x)=-f(x)</math>.

Every y-value changes sign, so a point <math>(x,y)</math> becomes <math>(x,-y)</math>.

For a reflection across the '''y-axis''':
<math>g(x)=f(-x)</math>.

Every x-value changes sign, so a point <math>(x,y)</math> becomes <math>(-x,y)</math>.

For example, the graph of <math>y=-x^2</math> is the reflection of <math>y=x^2</math> across the x-axis. It opens downward instead of upward.

{{#ev:youtube|https://www.youtube.com/watch?v=mR2y_kohZIw|500|center}}

{{BR}}
= Vertical Stretches and Compressions =

If
<math>g(x)=a\,f(x)</math>,

then every output value is multiplied by <math>a</math>.

When <math>|a|>1</math>, the graph is stretched vertically. Points move farther from the x-axis.

When <math>0<|a|<1</math>, the graph is compressed vertically. Points move closer to the x-axis.

When <math>a<0</math>, the graph is also reflected across the x-axis.

For the quadratic parent <math>f(x)=x^2</math>:
# <math>g(x)=3x^2</math> is narrower because of a vertical stretch by a factor of three.
# <math>g(x)=\frac{1}{2}x^2</math> is wider because of a vertical compression by a factor of one half.
# <math>g(x)=-2x^2</math> is vertically stretched by a factor of two and reflected across the x-axis.

[[File:Family of parabolas.svg|500px|frameless|center]]

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

{{#ev:youtube|https://www.youtube.com/watch?v=1lt6a77OmLU|500|center}}

{{BR}}
= Horizontal Stretches and Compressions =

If
<math>g(x)=f(bx)</math>,

then x-coordinates are divided by <math>b</math>.

For positive <math>b</math>:
# If <math>b>1</math>, the graph is compressed horizontally by a factor of <math>\frac{1}{b}</math>.
# If <math>0<b<1</math>, the graph is stretched horizontally by a factor of <math>\frac{1}{b}</math>.

If <math>b<0</math>, there is also a reflection across the y-axis.

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

{{#ev:youtube|https://www.youtube.com/watch?v=npMMbXm9sR0|500|center}}

{{BR}}
= A General Transformation Rule =

Many transformations can be organized with the form

<math>g(x)=a\,f(b(x-h))+k</math>.

Each parameter has a job:

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

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

<math>\left(h+\frac{u}{b},\;k+av\right)</math>,

provided <math>b\neq0</math>.

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

{{BR}}
= Quadratic Functions in Vertex Form =

A quadratic function in vertex form is

<math>y=a(x-h)^2+k</math>.

It is a transformation of the parent function <math>y=x^2</math>.

You can read several features immediately:
# The vertex is <math>(h,k)</math>.
# The axis of symmetry is <math>x=h</math>.
# If <math>a>0</math>, the parabola opens upward.
# If <math>a<0</math>, the parabola opens downward.
# If <math>|a|>1</math>, the parabola is narrower than <math>y=x^2</math>.
# If <math>0<|a|<1</math>, the parabola is wider than <math>y=x^2</math>.

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

{{BR}}
= Worked Examples =

'''Example 1: Translation and vertical stretch'''

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

'''Example 2: Absolute value with reflection'''

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

'''Example 3: Factoring the input first'''

Suppose <math>g(x)=f(2x-4)+1</math>. First rewrite the input as <math>2(x-2)</math>. 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:

{{#ev:youtube|https://www.youtube.com/watch?v=sTCRB6hMsC4|500|center}}

{{BR}}
= 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.

{{BR}}
= 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.

{{BR}}
= Common Mistakes and How to Avoid Them =

'''Mistake 1: Reversing horizontal shift signs.''' Remember that <math>f(x-h)</math> moves right when <math>h</math> 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 <math>f(2x-4)</math>, rewrite it as <math>f(2(x-2))</math> 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.

{{BR}}
= Interactive Tasks =

{{BR}}
== Quiz: Test Your Knowledge ==

{{MC}}

'''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)

{{E}}
<br>

{{MC}}

'''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)

{{E}}
<br>

{{MC}}

'''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)

{{E}}
<br>

{{MC}}

'''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)

{{E}}
<br>

{{MC}}

'''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)

{{E}}
<br>

{{MC}}

'''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)

{{E}}
<br>

{{MC}}

'''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)

{{E}}
<br>

{{MC}}

'''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)

{{E}}
<br>

{{MC}}

'''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)

{{E}}
<br>

{{MC}}

'''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)

{{E}}
<br>

{{BR}}
== Memory Game ==

<div class="memo-quiz">
{|
|-
| 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
|}
{{E}}
<br>

{{BR}}
== Drag and Drop ==

<div class="lueckentext-quiz">

{| class="wikitable"
! 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
|}
{{E}}

<br>
...
<br />

{{BR}}
== Crossword Puzzle ==

<div class="kreuzwort-quiz">
{|
|-
| 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?
|}
{{E}}
<br>

{{BR}}
== LearningApps ==

<iframe> https://learningapps.org/index.php?s=Transformations+of+Functions </iframe>

{{BR}}
== Cloze Text ==

<quiz display=simple>
{'''Complete the text.'''<br>
|type="{}"}
A parent function gives the { basic shape } of a family of related graphs. Adding a constant outside a function creates a { vertical translation }. Replacing x with x minus h produces a shift to the { right } when h is positive. A negative sign outside the function creates an { x-axis reflection }. A negative sign on the input creates a { y-axis reflection }. Multiplying all outputs by a factor greater than one creates a { vertical stretch }. In the expression f(bx), x-coordinates are scaled by the reciprocal of { b }. For a quadratic in vertex form, the point h comma k is the { vertex }.
</quiz>

<br>

{{BR}}
= Open-Ended Tasks =

{{BR}}
=== Easy ===
# [[English:Graph Sketch Gallery|Graph Sketch Gallery]]: Draw a parent function and three transformed versions of it, label each equation, and write one sentence explaining every change.
# [[English:Transformation Vocabulary Cards|Transformation Vocabulary Cards]]: Create a set of illustrated cards for translation, reflection, stretch, and compression, using one original example on each card.
# [[English:Digital Slider Experiment|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.
# [[English:Mini Explainer Video|Mini Explainer Video]]: Record a short video in which you teach the difference between an inside change and an outside change in function notation.

{{BR}}
=== Standard ===
# [[English:Quadratic Design Challenge|Quadratic Design Challenge]]: Design three parabolas from <math>y=x^2</math> using different values of a, h, and k, then explain how each parameter affects the graph.
# [[English:Peer Transformation Interview|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.
# [[English:Graph Error Detective|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.
# [[English:Motion Model Experiment|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.

{{BR}}
=== Advanced ===
# [[English:Transformation Composition Investigation|Transformation Composition Investigation]]: Compare two different orders of transformations on the same parent function and determine which operations commute and which do not.
# [[English:Real-World Modeling Project|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.
# [[English:Point Mapping Proof|Point Mapping Proof]]: Use the general form <math>g(x)=a f(b(x-h))+k</math> to derive the point-mapping rule and verify it with at least three parent-function points.
# [[English:Mathematical Photo Trail|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.

{{:Open Task - Create a MOOC}}

{{BR}}
= Learning Assessment =

# [[English:Transformation Reasoning|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.
# [[English:Equation Construction|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.
# [[English:Error Analysis|Error Analysis]]: Analyze a solution that claims <math>f(x+4)</math> shifts a graph right four units, identify the misconception, and correct the reasoning with a point example.
# [[English:Feature Transfer|Feature Transfer]]: Predict how the domain, range, intercepts, and symmetry of a parent function change under a specified transformation without graphing every point.
# [[English:Model Comparison|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.
# [[English:General Rule Application|General Rule Application]]: Use the point-mapping rule for <math>g(x)=a f(b(x-h))+k</math> to transform several key points and check the result against the transformed equation.

<br>
<br>

{{BR}}
= 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.

<br>
<br>

{{BR}}
= 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.

<iframe> https://en.m.wikipedia.org/wiki/Graph_of_a_function </iframe>

<br>

{{BR}}
= Linked Learning Areas =

{| align=center
{{:D-Tab}}
'''[[English:Transformations of Functions|Transformations of Functions]]'''
# [[English:Function|Function]]
# [[English:Graph of a function|Graph of a function]]
# [[English:Coordinate system|Coordinate system]]
# [[English:Linear function|Linear function]]
# [[English:Quadratic function|Quadratic function]]
# [[English:Absolute value|Absolute value]]
# [[English:Cubic function|Cubic function]]
# [[English:Square root|Square root]]
# [[English:Translation|Translation]]
# [[English:Reflection|Reflection]]
# [[English:Scaling|Scaling]]
# [[English:Vertex|Vertex]]
# [[English:Domain of a function|Domain of a function]]
# [[English:Range of a function|Range of a function]]
# [[English:Mathematical modeling|Mathematical modeling]]
|}

[[Category:English]]
[[Category:Transformations of Functions]]
[[Category:Mathematics]]
[[Category:Algebra]]
[[Category:Functions]]
[[Category:Graphing]]
[[Category:Grades 9-10]]
[[Category:Secondary Education]]

{{BR}}
= aiMOOC Projects =
[[Category:English]]
[[Category:Transformations of Functions]]
[[Category:Mathematics]]
[[Category:Algebra]]
[[Category:Functions]]
[[Category:Graphing]]
[[Category:Grades 9-10]]
[[Category:Secondary Education]]
[[Category:AI_MOOC]]
[[Category:GPT aiMOOC]]
{{MT}}

English:Transformations of Functions - Versionsgeschichte

Glanz: aiMOOC über GPT aiMOOC Action erstellt