Transformation of functions

Transformation of functions

Professor John drew the graph of the function \(f_{1}(x)=x^2\) on the board.

A professor drew the graph of the quadratic function on the board

The students drew this graph in their note books.

The professor then asked the students to draw the graph of \(f_{2}(x)=x^2+3\).

Do you think the students would have to draw this graph from scratch?

Do you think there is an easier way to indicate the graph for this function?

Students will not have to create it all over again because \(f_{2}(x)=x^2+3\) is the transformation of \(f_{1}(x)=x^2\).

In this mini-lesson, we will learn about transformations of functions, transformations of exponential functions, transformations of functions worksheet, and transformations of quadratic functions in the concept of transformation of the functions. Check out the interactive simulations to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

Lesson Plan

What Is the Meaning of Transformations of Functions?

The word "transform" means "to change from one form to another."

Transformations of functions mean transforming the function from one form to another.

Transformation of functions is a unique way of changing the formula of a function minimally and playing around with the graph.

Look at the graph of the function \(f(x)=x^2+3\).

graph of the quadratic function

Did you observe that the graph is 3 units above the quadratic function \(g(x)=x^2\)?

Visualize this transformation more clearly by drawing both the functions on the same coordinate plane.

Transformation on graph of the function


What Are the Types of Transformations of Functions?

The four common types of transformations are translation, rotation, reflection, and dilation.

Transformation Function
Rotation Rotates or turns the curve around an axis.
Reflection Flips the curve and produces the mirror-image.
Translation Slides or moves the curve.
Dilation Stretches or shrinks the curve.

Here is a simulation for you to experiment with all types of transformations.


How To Make Transformations of a Function?

Here are the rules of transformations of functions that could be applied to the graphs of functions.

Transformations of Quadratic Functions

We can apply the transformation rules to graphs of functions.

  • Here is the graph of a function that shows the transformation of reflection.

The red curve shows the graph of the function \(f(x) = x^3\).

The transformation \(g(x) =- x^3\) is done and it fetches the reflection of \(f(x)\) about the \(x\)-axis.

transformation - reflection graph

  • Now, let us observe the transformation of translation.

This red curve shows the graph of the function \(f(x) = x^2\)

The transformation of quadratic function \(f(x) = (x+2)^2\) shifts the parabola 2 steps right.

parabola-transformation-translation

  • In the function graph below, we observe the transformation of rotation.

To rotate 90\(^\circ\): \((x,y)\rightarrow (-y, x)\)

To rotate 180\(^\circ\): \((x,y) \rightarrow (-x,-y)\)

To rotate 270\(^\circ\): \((x,y) \rightarrow (y, -x)\)

Here we see that the preimage is rotated to 180\(^\circ\).

transformation of rotation

Want to know a general way to transform a function?

Here are some tricks for you to transform the given graph of the function.

 
tips and tricks
Tips and Tricks
  1. \(f(x+c)\) horizontally shifts the graph of \(f(x)\) left by \(c\) units.
  2. \(f(x-c)\) horizontally shifts the graph of \(f(x)\) right by \(c\) units.
  3. \(f(x)+c\) vertically shifts the graph of \(f(x)\) upward by \(c\) units.
  4. \(f(x)-c\) vertically shifts the graph of \(f(x)\) downward by \(c\) units.
  5. \(cf(x)\) vertically stretches the graph of \(f(x)\) by a factor of \(c\) units.
  6. \(\dfrac{1}{c}f(x)\) vertically shrinks the graph of \(f(x)\) by a factor of \(c\) units.
  7. \(f(cx)\) horizontally shrinks the graph of \(f(x)\) by a factor of \(c\) units.
  8. \(f\left(\dfrac{x}{c}\right)\) horizontally stretches the graph of \(f(x)\) by a factor of \(c\) units.
  9. \(-f(x)\) reflects the graph of \(f(x)\) over the \(x\)-axis.
  10. \(f(-x)\) reflects the graph of \(f(x)\) over the \(y\)-axis.

Solved Examples

Example 1

 

 

How has the below figure transformed?

Transformation of a shape

Solution

The rotation of an image turns its position around an axis.

Thus, here we can observe that the figure is rotated at \(90^{\circ}\) clockwise.

Thus, the transformation of rotation can be observed in the figure.
Example 2

 

 

The following graph shows the basic function \(y=b^x\).

transformation of exponential functions

Use the transformation of exponential functions to show the horizontal stretch and the reflection over the \(x\)-axis to draw the graph of the function \(h(x)=-2^{\frac{1}{4}}\).

Solution

We already know the curve of the basic function \(y=b^x\), where \(b=2\).

To stretch the graph horizontally by a factor of 4, we will graph the function \(f(x)=2^{\frac{x}{4}}\).

To show the reflection over the \(x\)-axis, we will graph the function \(h(x)=-2^{\frac{x}{4}}\).

transformation of exponential functions to show the horizontal stretch and the reflection over the x axis

This shows the transformation of exponential function \(y=2^x\).

So, the graph of the function \(h(x)=-2^{\frac{1}{4}}\) is shown.
Example 3

 

 

What do the following transformations do to the graph?

i) \(f(x)\rightarrow f(x)-3\)

ii) \(f(x)\rightarrow f(x-3)\)

Solution

\(f(x)\rightarrow f(x)-3\)

The \(y\)-coordinate undergoes the change by 3 units.

Thus, the transformation here is translation 3 units down.

\(f(x)\rightarrow f(x-3)\)

The \(x\)-coordinate undergoes the change of 3 units.

Thus, the transformation here is translation 3 units right.

 \(\therefore\) i) Translation is 3 units down
ii) Translation is 3 units right
 
Thinking out of the box
Think Tank
  1. Can you predict what do the following transformations do to the graph?
    (i)  \(f(x)\rightarrow f(x-5)+5\)
    (ii)  \(f(x)\rightarrow f(3x+1)-1\)
  2. How has the line \(y=x\) transformed to the line \(y=-x\)?

Interactive Questions

Here are a few activities for you to practice.

Select/type your answer and click the "Check Answer" button to see the result.

 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of transformation functions. The math journey around the transformation of functions starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions (FAQs)

1. How to do transformations of functions?

Follow the rules of the different transformations to find the transformations of functions.

For example, \(f(x+c)\) horizontally shifts the graph of \(f(x)\) left by \(c\) units.

2. How to describe transformations of functions?

Use the following points to describe the different transformations.

  • \(f(x+c)\) horizontally shifts the graph of \(f(x)\) left by \(c\) units.
  • \(f(x-c)\) horizontally shifts the graph of \(f(x)\) right by \(c\) units.
  • \(f(x)+c\) vertically shifts the graph of \(f(x)\) upward by \(c\) units.
  • \(f(x)-c\) vertically shifts the graph of \(f(x)\) downward by \(c\) units.
  • \(cf(x)\) vertically stretches the graph of \(f(x)\) by a factor of \(c\) units.
  • \(\dfrac{1}{c}f(x)\) vertically shrinks the graph of \(f(x)\) by a factor of \(c\) units.
  • \(f(cx)\) horizontally shrinks the graph of \(f(x)\) by a factor of \(c\) units.
  • \(f\left(\dfrac{x}{c}\right)\) horizontally stretches the graph of \(f(x)\) by a factor of \(c\) units.
  • \(-f(x)\) reflects the graph of \(f(x)\) over the \(x\)-axis.
  • \(f(-x)\) reflects the graph of \(f(x)\) over the \(y\)-axis.

3. How to find the transformations of functions?

Follow the rules of the different transformations to find the transformations of functions.

For example, \(f(x+c)\) horizontally shifts the graph of \(f(x)\) left by \(c\) units.

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More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus
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