StudySmarter - The all-in-one study app.

4.8 • +11k Ratings

More than 3 Million Downloads

Free

Suggested languages for you:

Americas

Europe

Function Transformations

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Area Between Two Curves
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits at Infinity and Asymptotes
- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Radius of Convergence
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Separation of Variables
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Vectors in Space
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Conservation of Mechanical Energy
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Elastic Strings and Springs
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Power
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Work Done by a Constant Force
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Argand Diagram
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- De Moivre's Theorem
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Roots of Unity
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Bias in Experiments
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
- Conducting a Survey
- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
- Correlation Math
- Cumulative Distribution Function
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Degrees of Freedom
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Estimator Bias
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Test for Regression Slope
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Normal Distribution Percentile
- Paired T-Test
- Point Estimation
- Probability
- Probability Calculations
- Probability Density Function
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Quartiles
- Random Variables
- Randomized Block Design
- Residual Sum of Squares
- Residuals
- Sample Mean
- Sample Proportion
- Sampling
- Sampling Distribution
- Scatter Graphs
- Single Variable Data
- Skewness
- Spearman's Rank Correlation Coefficient
- Standard Deviation
- Standard Error
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Sum of Independent Random Variables
- Survey Bias
- T-distribution
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Variance for Binomial Distribution
- Venn Diagrams

You wake up in the morning, lazily stroll to the bathroom, and still half-asleep you start combing your hair – after all, style first. On the other side of the mirror, your image, looking just as tired as you do, is doing the same – but she's holding the comb in the other hand. **What the hell is going on?**

Your image is being transformed by the mirror – more precisely, it's being **reflected. **Transformations like this happen every day and every morning in our world, as well as in the much less chaotic and confusing world of Calculus.

Throughout calculus, you will be asked to **transform** and **translate** functions. What does this mean, exactly? It means taking one function and applying changes to it to create a new function. This is how graphs of functions can be transformed into different ones to represent different functions!

In this article, you will explore function transformations, their rules, some common mistakes, and cover plenty of examples!

It'd be a good idea to have a good grasp of the general concepts of various types of functions before taking a dive into this article: make sure to first read the article on Functions!

- Function transformations: meaning
- Function transformations: rules
- Function transformations: common mistakes
- Function transformations: order of operations
- Function transformations: transformations of a point
- Function transformations: examples

So, what are function transformations? So far, you have learned about **parent functions** and how their function families share a similar shape. You can further your knowledge by learning how to transform functions.

**Function transformations** are the processes used on an existing function and its graph to give you a modified version of that function and its graph that has a similar shape to the original function.

When transforming a function, you should usually refer to the parent function to describe the transformations performed. However, depending on the situation, you might want to refer to the original function that was given to describe the changes.

Examples of a parent function (blue) and some of its possible transformations (green, pink, purple).

As illustrated by the image above, function transformations come in various forms and affect the graphs in different ways. That being said, we can break down the transformations into **two major categories**:

**Horizontal**transformations**Vertical**transformations

**Any function can be transformed**, horizontally and/or vertically, via** four main types of transformations**:

Horizontal and vertical

**shifts**(or translations)Horizontal and vertical

**shrinks**(or compressions)Horizontal and vertical

**stretches**Horizontal and vertical

**reflections**

Horizontal transformations only change the \(x\)-coordinates of functions. Vertical transformations only change the \(y\)-coordinates of functions.

You can use a table to summarize the different transformations and their corresponding effects on the graph of a function.

Transformation of \( f(x) \), where \( c > 0 \) | Effect on the graph of \( f(x) \) |

\( f(x)+c \) | Vertical shift up by \(c\) units |

\( f(x)-c \) | Vertical shift down by \(c\) units |

\( f(x+c) \) | Horizontal shift left by \(c\) units |

\( f(x-c) \) | Horizontal shift right by \(c\) units |

\( c \left( f(x) \right) \) | Vertical stretch by \(c\) units, if \( c > 1 \)Vertical shrink by \(c\) units, if \( 0 < c < 1 \) |

\( f(cx) \) | Horizontal stretch by \(c\) units, if \( 0 < c < 1 \)Horizontal shrink by \(c\) units, if \( c > 1 \) |

\( -f(x) \) | Vertical reflection (over the \(\bf{x}\)-axis) |

\( f(-x) \) | Horizontal reflection (over the \(\bf{y}\)-axis) |

**Horizontal** transformations are made when you act on a **function's input variable** (usually \(x\)). You can

add or subtract a number from the function's input variable, or

multiply the function's input variable by a number.

Here is a summary of how horizontal transformations work:

**Shifts**– Adding a number to \(x\) shifts the function to the left; subtracting shifts it to the right.**Shrinks**– Multiplying \(x\) by a number whose magnitude is greater than \(1\)**shrinks**the function horizontally.**Stretches**– Multiplying \(x\) by a number whose magnitude is less than \(1\)**stretches**the function horizontally.**Reflections**– Multiplying \(x\) by \(-1\) reflects the function horizontally (over the \(y\)-axis).

Horizontal transformations, except reflection, **work the opposite way you'd expect them to!**

Consider the parent function from the image above:

\[ f(x) = x^{2} \]

This is the parent function of a parabola. Now, say you want to transform this function by:

- Shifting it to the left by \(5\) units
- Shrinking it horizontally by a factor of \(2\)
- Reflecting it over the \(y\)-axis

How can you do that?

**Solution**:

- Graph the parent function.
- Write the transformed function.
- Start with the parent function:
- \( f_{0}(x) = x^{2} \)

- Add in the shift to the left by \(5\) units by putting parentheses around the input variable, \(x\), and putting \(+5\) within those parentheses after the \(x\):
- \( f_{1}(x) = f_{0}(x+5) = \left( x+5 \right)^{2} \)

- Next, multiply the \(x\) by \(2\) to shrink it horizontally:
- \( f_{2}(x) = f_{1}(2x) = \left( 2x+5 \right)^{2} \)

- Finally, to reflect over the \(y\)-axis, multiply \(x\) by \(-1\):
- \( f_{3}(x) = f_{2}(-x) = \left( -2x+5 \right)^{2} \)

- So, your final transformed function is:
- \( \bf{ f(x) } = \bf{ \left( -2x + 5 \right)^{2} } \)

- Start with the parent function:
- Graph the transformed function, and compare it to the parent to make sure the transformations make sense.
- Things to note here:
- The transformed function is on the right due to the \(y\)-axis reflection performed after the shift.
- The transformed function is shifted by \(2.5\) instead of \(5\) due to the shrinking by a factor of \(2\).

**Vertical** transformations are made when you act on the **entire function.** You can either

add or subtract a number from the entire function, or

multiply the entire function

Unlike horizontal transformations, vertical transformations work the way you expect them to (yay!). Here is a summary of how vertical transformations work:

**Shifts**– Adding a number to the entire function shifts it up; subtracting shifts it down.**Shrinks**– Multiplying the entire function by a number whose magnitude is less than \(1\)**shrinks**the function.**Stretches**– Multiplying the entire function by a number whose magnitude is greater than \(1\)**stretches**the function.**Reflections**– Multiplying the entire function by \(-1\) reflects it vertically (over the \(x\)-axis).

Again, consider the parent function:

\[ f(x) = x^{2} \]

Now, say you want to transform this function by

- Shifting it up by \(5\) units
- Shrinking it vertically by a factor of \(2\)
- Reflecting it over the \(x\)-axis

How can you do that?

**Solution**:

- Graph the parent function.
- Write the transformed function.
- Start with the parent function:
- \( f_{0}(x) = x^{2} \)

- Add in the shift up by \(5\) units by putting \(+5\) after \( x^{2} \):
- \( f_{1}(x) = f_{0}(x) + 5 = x^{2} + 5 \)

- Next, multiply the function by \( \frac{1}{2} \) to compress it vertically by a factor of \(2\):
- \( f_{2}(x) = \frac{1}{2} \left( f_{1}(x) \right) = \frac{x^{2}+5}{2} \)

- Finally, to reflect over the \(x\)-axis, multiply the function by \(-1\):
- \( f_{3}(x) = -f_{2}(x) = - \frac{x^{2}+5}{2} \)

- So, your final transformed function is:
- \( \bf{ f(x) } = \bf{ - \frac{x^{2}+5}{2} } \)

- Start with the parent function:
- Graph the transformed function, and compare it to the parent to make sure the transformations make sense.

It is tempting to think that the horizontal transformation of adding to the independent variable, \(x\), moves the function's graph to the right because you think of adding as moving to the right on a number line. This, however, is not the case.

Remember, **horizontal transformations** move the graph the **opposite **way you expect them to!

Let's say you have the function, \( f(x) \), and its transformation, \( f(x+3) \). How does the \(+3\) move the graph of \( f(x) \)?

**Solution**:

- This is a
**horizontal transformation**because the addition is applied to the independent variable, \(x\).- Therefore, you know that the
**graph****moves opposite to what you'd expect**.

- Therefore, you know that the
- The graph of \( f(x) \) is moved to the
**left by 3 units**.

If horizontal transforms are still a bit confusing, consider this.

Look at the function, \( f(x) \), and its transformation, \( f(x+3) \), again and think about the point on the graph of \( f(x) \) where \( x = 0 \). So, you have \( f(0) \) for the original function.

- What does \(x\) need to be in the transformed function so that \( f(x+3) = f(0) \)?
- In this case, \(x\) needs to be \(-3\).
- So, you get: \( f(-3+3) = f(0) \).
- This means you need to
**shift the graph left by 3 units**, which makes sense with what you think of when you see a negative number.

When identifying whether a transformation is horizontal or vertical, keep in mind that **transformations are only horizontal if they are applied to \(x\) when it has a power of \(1\)**.

Consider the functions:

\[ g(x) = x^{3} - 4 \]

and

\[ h(x) = (x-4)^{3} \]

Take a minute to think about how these two functions, with respect to their parent function \( f(x) = x^{3} \), are transformed.

Can you compare and contrast their transformations? What do their graphs look like?

**Solution**:

- Graph the parent function.
- Determine the transformations indicated by the \( g(x) \) and \( h(x) \).
- For \( g(x) \):
- Since \(4\) is subtracted from the entire function, not just the input variable \(x\), the graph of \( g(x) \) shifts vertically down by \(4\) units.

- For \( h(x) \):
- Since \(4\) is subtracted from the input variable \(x\), not the entire function, the graph of \( h(x) \) shifts horizontally to the right by \(4\) units.

- For \( g(x) \):
- Graph the transformed functions with the parent function and compare them.

Let's look at another common mistake.

Expanding on the previous example, now consider the function:

\[ f(x) = \frac{1}{2} \left( x^{3} - 4 \right) + 2 \]

At first glance, you might think this has a horizontal shift of \(4\) units with respect to the parent function \( f(x) = x^{3} \).

This is not the case!

While you might be tempted to think so due to the parentheses, the \( \left( x^{3} - 4 \right) \) **does not indicate a horizontal shift** because \(x\) has a power of \(3\), not \(1\). Therefore, \( \left( x^{3} - 4 \right) \) **indicates a vertical shift** of \(4\) units down with respect to the parent function \( f(x) = x^{3} \).

To get the complete translation information, you must expand and simplify:

\[ \begin{align}f(x) &= \frac{1}{2} \left( x^{3} - 4 \right) + 2 \\&= \frac{1}{2} x^{3} - 2 + 2 \\&= \frac{1}{2} x^{3}\end{align} \]

This tells you that there is, in fact, no vertical or horizontal translation. There is only a vertical compression by a factor of \(2\)!

Let's compare this function to one that looks very similar but is transformed much differently.

\( f(x) = \frac{1}{2} \left( x^{3} - 4 \right) + 2 = \frac{1}{2} x^{3} \) | \( f(x) = \frac{1}{2} (x - 4)^{3} + 2 \) |

vertical compression by a factor of \(2\) | vertical compression by a factor of \(2\) |

no horizontal or vertical translation | horizontal translation \(4\) units right |

vertical translation \(2\) units up |

You have to ensure the coefficient of the \(x\) term is factored out fully to get an accurate analysis of the horizontal translation.

Consider the function:

\[ g(x) = 2(3x + 12)^{2} +1 \]

At first glance, you might think this function is shifted \(12\) units to the left with respect to its parent function, \( f(x) = x^{2} \).

This is not the case! While you might be tempted to think so due to the parentheses, the \( (3x + 12)^{2} \) does not indicate a left shift of \(12\) units. You must factor out the coefficient on \(x\)!

\[ g(x) = 2(3(x + 4)^{2}) + 1 \]

Here, you can see that the function is actually shifted \(4\) units left, not \(12\), after writing the equation in the proper form. The graph below serves to prove this.

.

As with most things in math, the **order** in which transformations of functions are done matters. For instance, considering the parent function of a parabola,

\[ f(x) = x^{2} \]

If you were to apply a vertical stretch of \(3\) and then a vertical shift of \(2\), you would get a **different final graph** than if you were to apply a vertical shift of \(2\) and then a vertical stretch of \(3\). In other words,

\[ \begin{align}2 + 3f(x) &\neq 3(2 + f(x)) \\2 + 3(x^{2}) &\neq 3(2 + x^{2})\end{align} \]

The table below visualizes this.

A vertical stretch of \(3\), then a vertical shift of \(2\) | A vertical shift of \(2\), then a vertical stretch of \(3\) |

And as with most rules, there are exceptions! There are situations where the order doesn't matter, and the same transformed graph will be generated regardless of the order in which the transformations are applied.

The order of transformations **matters **when

there are transformations within the

**same category**(i.e., horizontal or vertical)but are

**not the same type**(i.e., shifts, shrinks, stretches, compressions).

What does this mean? Well, look the example above again.

Do you notice how the transformation (green) of the parent function (blue) looks quite different between the two images?

That is because the transformations of the parent function were the **same category** (i.e., **vertical **transformation), but were a **different type** (i.e., a **stretch **and a **shift**). If you change the order in which you perform these transformations, you get a different result!

So, to generalize this concept:

Say you want to perform \( 2 \) different horizontal transformations on a function:

No matter which \( 2 \) types of horizontal transformations you choose, if they are not the same (e.g., \( 2 \) horizontal shifts), the order in which you apply these transforms matters.

Say you want to perform \( 2 \) different vertical transformations on another function:

No matter which \( 2 \) types of vertical transformations you choose, if they are not the same (e.g., \( 2 \) vertical shifts), the order in which you apply these transforms matters.

Function transformations of the **same category**, but **different types** **do not commute** (i.e., the **order matters**).

Say you have a function, \( f_{0}(x) \), and constants \( a \) and \( b \).

Looking at horizontal transformations:

- Say you want to apply a horizontal shift and a horizontal stretch (or shrink) to a general function. Then, if you apply the horizontal stretch (or shrink) first, you get:\[ \begin{align}f_{1}(x) &= f_{0}(ax) \\f_{2}(x) &= f_{1}(x+b) = f_{0} \left( a(x+b) \right)\end{align} \]
- Now, if you apply the horizontal shift first, you get:\[ \begin{align}g_{1}(x) &= f_{0}(x+b) \\g_{2}(x) &= g_{1}(ax) = f_{0}(ax+b)\end{align} \]
- When you compare these two results, you see that:\[ \begin{align}f_{2}(x) &\neq g_{2}(x) \\f_{0} \left( a(x+b) \right) &\neq f_{0}(ax+b)\end{align} \]

Looking at vertical transformations:

- Say you want to apply a vertical shift and a vertical stretch (or shrink) to a general function. Then, if you apply the vertical stretch (or shrink) first, you get:\[ \begin{align}f_{1}(x) &= af_{0}(x) \\f_{2}(x) &= b+f_{1}(x) = b+af_{0}(x)\end{align} \]
- Now, if you apply the vertical shift first, you get:\[ \begin{align}g_{1}(x) &= b+f_{0}(x) \\g_{2}(x) &= ag_{1}(x) = a \left( b+f_{0}(x) \right)\end{align} \]
- When you compare these two results, you see that:\[ \begin{align}f_{2}(x) &\neq g_{2}(x) \\b+af_{0}(x) &\neq a \left( b+f_{0}(x) \right)\end{align} \]

The order of transformations **does not matter** when

- there are transformations within the
**same category**and are the**same type**, or - there are transformations that are
**different categories**altogether.

What does this mean?

If you have a function that you want to apply multiple transformations of the same category and type, the order does not matter.

You can apply horizontal stretches/shrinks in any order and get the same result.

You can apply horizontal shifts in any order and get the same result.

You can apply horizontal reflections in any order and get the same result.

You can apply vertical stretches/shrinks in any order and get the same result.

You can apply vertical shifts in any order and get the same result.

You can apply vertical reflections in any order and get the same result.

If you have a function that you want to apply transformations of different categories, the order does not matter.

You can apply a horizontal and a vertical transformation in any order and get the same result.

Function transformations of the **same category** and **same type** **do commute** (i.e., the **order does not matter**).

Say you have a function, \( f_{0}(x) \), and constants \( a \) and \( b \).

- If you want to apply multiple horizontal stretches/shrinks, you get:\[ \begin{align}f_{1}(x) &= f_{0}(ax) \\f_{2}(x) &= f_{1}(bx) \\&= f_{0}(abx)\end{align} \]
- The product \(ab\) is commutative, so the order of the two horizontal stretches/shrinks does not matter.

- If you want to apply multiple horizontal shifts, you get:\[ \begin{align}f_{1}(x) &= f_{0}(a+x) \\f_{2}(x) &= f_{1}(b+x) \\&= f_{0}(a+b+x)\end{align} \]
- The sum \(a+b\) is commutative, so the order of the two horizontal shifts does not matter.

- If you want to apply multiple vertical stretches/shrinks, you get:\[ \begin{align}f_{1}(x) &= af_{0}(x) \\f_{2}(x) &= bf_{1}(x) \\&= abf_{0}(x)\end{align} \]
- The product \(ab\) is commutative, so the order of the two vertical stretches/shrinks does not matter.

- If you want to apply multiple vertical shifts, you get:\[ \begin{align}f_{1}(x) &= a + f_{0}(x) \\f_{2}(x) &= b + f_{1}(x) \\&= a + b + f_{0}(x)\end{align} \]
- The sum \(a+b\) is commutative, so the order of the two vertical shifts does not matter.

Let's look at another example.

Function transformations that are **different categories** **do commute** (i.e., the **order does not matter**).

Say you have a function, \( f_{0}(x) \), and constants \( a \) and \( b \).

- If you want to combine a horizontal stretch/shrink and a vertical stretch/shrink, you get:\[ \begin{align}f_{1}(x) &= f_{0}(ax) \\f_{2}(x) &= bf_{1}(x) \\&= bf_{0}(ax)\end{align} \]
- Now, if you reverse the order in which these two transformations are applied, you get:\[ \begin{align}g_{1}(x) &= bf_{0}(x) \\g_{2}(x) &= g_{1}(ax) \\&= bf_{0}(ax)\end{align} \]
- When you compare these two results, you see that:\[ \begin{align}f_{2}(x) &= g_{2}(x) \\bf_{0}(ax) &= bf_{0}(ax)\end{align} \]

So, is there a **correct **order of operations when applying transformations to functions?

The short answer is no, you can apply transformations to functions in any order you wish to follow. As you saw in the common mistakes section, the trick is learning how to tell which transformations have been made, and in which order, when going from one function (usually a parent function) to another.

Now you are ready to transform some functions! To start, you will try to transform a point of a function. What you will do is move a specific point based on some given transformations.

If the point \( (2, -4) \) is on the function \( y = f(x) \), then what is the corresponding point on \( y = 2f(x-1)-3 \)?

**Solution**:

You know so far that the point \( (2, -4) \) is on the graph of \( y = f(x) \). So, you can say that:

\[ f(2) = -4 \]

What you need to find out is the corresponding point that is on \( y = 2f(x-1)-3 \). You do that by looking at the transformations given by this new function. Walking through these transformations, you get:

- Start with the parentheses.
- Here you have \( (x-1) \). → This means you shift the graph to the right by \(1\) unit.
- Since this is the only transformation applied to the input, you know there are no other horizontal transformations on the point.
- So, you know the
**transformed point has an \(x\)-coordinate of \(3\)**.

- So, you know the

- Apply the multiplication.
- Here you have \( 2f(x-1) \). → The \(2\) means you have a vertical stretch by a factor of \(2\), so your \(y\)-coordinate doubles to \(-8\).
- But, you are not done yet! You still have one more vertical transformation.

- Apply the addition/subtraction.
- Here you have the \(-3\) applied to the entire function. → This means you have a shift down, so you subtract \(3\) from your \(y\)-coordinate.
- So, you know the
**transformed point has a \(y\)-coordinate of \(-11\)**.

- So, you know the

- Here you have the \(-3\) applied to the entire function. → This means you have a shift down, so you subtract \(3\) from your \(y\)-coordinate.

So, with these transformations done to the function, whatever function it may be, the corresponding point to \( (2, -4) \) is the transformed point \( \bf{ (3, -11) } \).

To generalize this example, say you are given the function \( f(x) \), the point \( (x_0, f(x_0)) \), and the transformed function\[ g(y) = af(x = by+c)+d,\]what is the corresponding point?

First, you need to define what the corresponding point is:

It's the point on the transformed function's graph such that the \(x\)-coordinates of the original and the transformed point are related by the horizontal transformation.

So, you need to find the point \((y_0, g(y_0))\) such that

\[x_0 = by_0+c\]

To find \(y_0\), isolate it from the above equation:

\[y_0 = \frac{x_0-c}{b}\]

To find \(g(y_0)\), plug in \(g\):

\[g(y_0) = af(x = by_0+c)+d = af(x_0)+d\]

As in the example above, let \( (x_0, f(x_0)) = (2,-4) \), and\[a = 2, b = 1, c = -1, d = -3.\]So,\[y_0 = \frac{2-(-1)}{1} = 3, \quad g(y_0) = 2\cdot (-4) -3 = -11.\]

**Bottom line**: to find the \(x\)-component of the transformed point, solve the **inverted** horizontal transformation; to find the \(y\)-component of the transformed point, solve the vertical transformation.

Now let's look at some examples with different types of functions!

The general equation for a transformed exponential function is:

\[ f(x) = a(b)^{k(x-d)}+c \]

Where,

\[ a = \begin{cases}\mbox{vertical stretch if } a > 1, \\\mbox{vertical shrink if } 0 < a < 1, \\\mbox{reflection over } x-\mbox{axis if } a \mbox{ is negative}\end{cases} \]

\[ b = \mbox{the base of the exponential function} \]

\[ c = \begin{cases}\mbox{vertical shift up if } c \mbox{ is positive}, \\\mbox{vertical shift down if } c \mbox{ is negative}\end{cases} \]

\[ d = \begin{cases}\mbox{horizontal shift left if } +d \mbox{ is in parentheses}, \\\mbox{horizontal shift right if } -d \mbox{ is in parentheses}\end{cases} \]

\[ k = \begin{cases}\mbox{horizontal stretch if } 0 < k < 1, \\\mbox{horizontal shrink if } k > 1, \\\mbox{reflection over } y-\mbox{axis if } k \mbox{ is negative}\end{cases} \]

Let's transform the parent natural exponential function, \( f(x) = e^{x} \), by graphing the natural exponential function:

\[ f(x) = -e^{2(x-1)}+3. \]

**Solution**:

- Graph the parent function.
- Determine the transformations.
Start with the parentheses (horizontal shifts)

Here you have \(f(x) = e^{(x-1)}\), so the graph

**shifts to the right by \(1\) unit**.

Apply the multiplication (stretches and/or shrinks)

Here you have \( f(x) = e^{2(x-1)} \), so the graph

**shrinks horizontally by a factor of \(2\)**.

Apply the negations (reflections)

Here you have \( f(x) = -e^{2(x-1)} \), so the graph is

**reflected over the \(x\)-axis**.

Apply the addition/subtraction (vertical shifts)

Here you have \( f(x) = -e^{2(x-1)} + 3 \), so the

**graph is shifted up by \(3\) units**.

Graph the final transformed function.

The general equation for a transformed logarithmic function is:

\[ f(x) = a\mbox{log}_{b}(kx+d)+c. \]

Where,

\[ a = \begin{cases}\mbox{vertical stretch if } a > 1, \\\mbox{vertical shrink if } 0 < a < 1, \\\mbox{reflection over } x-\mbox{axis if } a \mbox{ is negative}\end{cases} \]

\[ b = \mbox{the base of the logarithmic function} \]

\[ c = \begin{cases}\mbox{vertical shift up if } c \mbox{ is positive}, \\\mbox{vertical shift down if } c \mbox{ is negative}\end{cases} \]

\[ d = \begin{cases}\mbox{horizontal shift left if } +d \mbox{ is in parentheses}, \\\mbox{horizontal shift right if } -d \mbox{ is in parentheses}\end{cases} \]

\[ k = \begin{cases}\mbox{horizontal stretch if } 0 < k < 1, \\\mbox{horizontal shrink if } k > 1, \\\mbox{reflection over } y-\mbox{axis if } k \mbox{ is negative}\end{cases} \]

Let's transform the parent natural log function, \( f(x) = \text{log}_{e}(x) = \text{ln}(x) \) by graphing the function:

\[ f(x) = -2\text{ln}(x+2)-3. \]

**Solution**:

- Graph the parent function.
- Determine the transformations.
Start with the parentheses (horizontal shifts)

Here you have \( f(x) = \text{ln}(x+2) \), so the

**graph shifts to the left by \(2\) units**.

Apply the multiplication (stretches and/or shrinks)

Here you have \( f(x) = 2\text{ln}(x+2) \), so the

**graph stretches vertically by a factor of \(2\)**.

Apply the negations (reflections)

Here you have \( f(x) = -2\text{ln}(x+2) \), so the

**graph reflects over the \(x\)-axis**.

Apply the addition/subtraction (vertical shifts)

Here you have \( f(x) = -2\text{ln}(x+2)-3 \), so the

**graph shifts down \(3\) units**.

- Graph the final transformed function.

The general equation for a rational function is:

\[ f(x) = \frac{P(x)}{Q(x)} ,\]

where

\[ P(x) \mbox{ and } Q(x) \mbox{ are polynomial functions, and } Q(x) \neq 0. \]

Since a rational function is made up of polynomial functions, the general equation for a transformed polynomial function applies to the numerator and denominator of a rational function. The general equation for a transformed polynomial function is:

\[ f(x) = a \left( f(k(x-d)) + c \right), \]

where,

\[ a = \begin{cases}\mbox{vertical stretch if } a > 1, \\\mbox{vertical shrink if } 0 < a < 1, \\\mbox{reflection over } x-\mbox{axis if } a \mbox{ is negative}\end{cases} \]

\[ c = \begin{cases}\mbox{vertical shift up if } c \mbox{ is positive}, \\\mbox{vertical shift down if } c \mbox{ is negative}\end{cases} \]

\[ d = \begin{cases}\mbox{horizontal shift left if } +d \mbox{ is in parentheses}, \\\mbox{horizontal shift right if } -d \mbox{ is in parentheses}\end{cases} \]

\[ k = \begin{cases}\mbox{horizontal stretch if } 0 < k < 1, \\\mbox{horizontal shrink if } k > 1, \\\mbox{reflection over } y-\mbox{axis if } k \mbox{ is negative}\end{cases} \]

Let's transform the parent reciprocal function, \( f(x) = \frac{1}{x} \) by graphing the function:

\[ f(x) = - \frac{2}{2x-6}+3. \]

**Solution**:

- Graph the parent function.
- Determine the transformations.
Start with the parentheses (horizontal shifts)

- To find the horizontal shifts of this function, you need to have the denominator in standard form (i.e., you need to factor out the coefficient of \(x\)).
- So, the transformed function becomes:\[ \begin{align}f(x) &= - \frac{2}{2x-6}+3 \\&= - \frac{2}{2(x-3)}+3\end{align} \]
- Now, you have \( f(x) = \frac{1}{x-3} \), so you know the
**graph shifts right by \(3\) units**.

Apply the multiplication (stretches and/or shrinks)

**This is a tricky step**Here you have a

**horizontal shrink by a factor of \(2\)**(from the \(2\) in the denominator) and a**vertical stretch by a factor of \(2\)**(from the \(2\) in the numerator).Here you have \( f(x) = \frac{2}{2(x-3)} \), which gives you the

**same graph**as \( f(x) = \frac{1}{x-3} \).- The graphs of the parent rational function (blue) and the first step of the transform (fucsia).

Apply the negations (reflections)

Here you have \( f(x) = - \frac{2}{2(x-3)} \), so the

**graph reflects over the \(x\)-axis**.- The graphs of the parent rational function (blue) and the first three steps of the transform (yellow, purple, pink).

Apply the addition/subtraction (vertical shifts)

Here you have \( f(x) = - \frac{2}{2(x-3)} + 3 \), so the

**graph shifts up \(3\) units**.

- Graph the final transformed function.
- The final transformed function is \( f(x) = - \frac{2}{2(x-3)} + 3 = - \frac{2}{2x-6} + 3 \).

**Function transformations**are the processes used on an existing function and its graph to give us a modified version of that function and its graph that has a similar shape to the original function.- Function transformations are broken down into
**two major categories**:Horizontal transformations

- Horizontal transformations are made when we either add/subtract a number from a function's input variable (usually x) or multiply it by a number.
**Horizontal transformations, except reflection, work in the opposite way we'd expect them to**. - Horizontal transformations only change the x-coordinates of functions.

- Horizontal transformations are made when we either add/subtract a number from a function's input variable (usually x) or multiply it by a number.
Vertical transformations

Vertical transformations are made when we either add/subtract a number from the entire function, or multiply the entire function by a number. Unlike horizontal transformations, vertical transformations work the way we expect them to.

- Vertical transformations only change y-coordinates of functions.

**Any function can be transformed**, horizontally and/or vertically, via**four main types of transformations**:Horizontal and vertical shifts (or translations)

Horizontal and vertical shrinks (or compressions)

Horizontal and vertical stretches

Horizontal and vertical reflections

- When identifying whether a transformation is horizontal or vertical, keep in mind that
**transformations are only horizontal if they are applied to x when it has a power of 1**.

The 4 transformations of a function are:

- Horizontal and vertical shifts (or translations)
- Horizontal and vertical shrinks (or compressions)
- Horizontal and vertical stretches
- Horizontal and vertical reflections

To find the transformation of a function at a point, follow these steps:

- Choose a point that lies on the function (or use a given point).
- Look for any Horizontal Transformations between the original function and the transformed function.
- Horizontal Transformations are what the x-value of the function is changed by.
- Horizontal Transformations only affect the x-coordinate of the point.
- Write the new x-coordinate.

- Look for any Vertical Transformations between the original function and the transformed function.
- Vertical Transformations are what the entire function is changed by.
- Vertical Transformation only affect the y-coordinate of the point.
- Write the new y-coordinate.

- With both the new x- and y-coordinates, you have the transformed point!

To graph an exponential function with transformations is the same process to graph any function with transformations.

Given an original function, say y = f(x), and a transformed function, say y = 2f(x-1)-3, let's graph the transformed function.

- Horizontal transformations are made when we either add/subtract a number from x, or multiply x by a number.
- In this case, the horizontal transformation is shifting the function to the right by 1.

- Vertical transformations are made when we either add/subtract a number from the entire function, or multiply the entire function by a number.
- In this case, the vertical transformations are:
- A vertical stretch by 2
- A vertical shift down by 3

- In this case, the vertical transformations are:
- With these transformations in mind, we now know that the graph of the transformed function is:
- Shifted to the right by 1 unit compared to the original function
- Shifted down by 3 units compared to the original function
- Stretched by 2 units compared to the original function

- To graph the function, simply choose input values of x and solve for y to get enough points to draw the graph.

^{2 }is y=3x^{2 }+5. This transformed equation undergoes a vertical stretch by a factor of 3 and a translation of 5 units up.

More about Function Transformations

Be perfectly prepared on time with an individual plan.

Test your knowledge with gamified quizzes.

Create and find flashcards in record time.

Create beautiful notes faster than ever before.

Have all your study materials in one place.

Upload unlimited documents and save them online.

Identify your study strength and weaknesses.

Set individual study goals and earn points reaching them.

Stop procrastinating with our study reminders.

Earn points, unlock badges and level up while studying.

Create flashcards in notes completely automatically.

Create the most beautiful study materials using our templates.

Sign up to highlight and take notes. It’s 100% free.

Over 10 million students from across the world are already learning smarter.

Get Started for Free