Function Transformations

Transformation of functions means that the curve representing the graph either "moves to left/right/up/down" or "it expands or compresses" or "it reflects". For example, the graph of the function f(x) = x² + 3 is obtained by just moving the graph of g(x) = x² by 3 units up. Function transformations are very helpful in graphing the functions just by moving/expanding/compressing/reflecting the curve without actually needing to graph it from scratch.

In this article, we will see what are the rules of function transformations and we will see how to do transformations of different types of functions along with examples.

A function transformation either "moves" or "resizes" or "reflects" the graph of the parent function. There are mainly three types of function transformations:

• Translation
• Dilation
• Reflection

Among these transformations, only dilation changes the size of the original shape but the other two transformations change the position of the shape but not the size of the shape. We can see what each of these transformations of functions mean in the table below.

In math words, the transformation of a function y = f(x) typically looks like y = a f(b(x + c)) + d. Here, a, b, c, and d are any real numbers and they represent transformations. Note that all outside numbers (that are outside the brackets) represent vertical transformations and all inside numbers represent horizontal transformations. Also, note that addition/subtraction indicates translation and multiplication/division represents dilation. Any minus sign multiplies means that it is a reflection. Here,

• 'a' represents the vertical dilation
• 'b' represents the horizontal dilation
• 'c' represents the horizontal translation
• 'd' represents the vertical translation

Let us learn each of these function transformations in detail.

A translation occurs when every point on a graph (representing a function) moves by the same amount in the same direction. There are two types of translations of functions.

• Horizontal translations
• Vertical translations

Horizontal Translation of Functions:

In this translation, the function moves to the left side or right side. This changes a function y = f(x) into the form y = f(x ± k), where 'k' represents the horizontal translation. Here,

• if k > 0, then the function moves to the left side by 'k' units.
• if k < 0, then the function moves to the right by 'k' units.

Here, the original function y = x² (y = f(x)) is moved to 3 units right to give the transformed function y = (x - 3)² (y = f(x - 3)).

Vertical Translation of Functions:

In this translation, the function moves to either up or down. This changes a function y = f(x) into the form f(x) ± k, where 'k' represents the vertical translation. Here,

• if k > 0, then the function moves up by 'k' units.
• if k < 0, then the function moves down by 'k' units.

Here, the original function y = x² (y = f(x)) is moved to 2 units up to give the transformed function y = x² + 2 (y = f(x) + 2).

A dilation is a stretch or a compression. If a graph undergoes dilation parallel to the x-axis, all the x-values are increased by the same scale factor. Similarly, if it is dilated parallel to the y-axis, all the y-values are increased by the same scale factor. There are two types of dilations.

• Horizontal Dilation
• Vertical Dilation

Horizontal Dilation

The horizontal dilation (also known as horizontal scaling) of a function either stretches/shrinks the curve horizontally. It changes a function y = f(x) into the form y = f(kx), with a scale factor '1/k', parallel to the x-axis. Here,

• If k > 1, then the graph shrinks.
• If 0 < k < 1, then the graph stretches.

In this dilation, there will be changes only in the x-coordinates but there won't be any changes in the y-coordinates. Every old x-coordinate is multiplied by 1/k to find the new x-coordinate. In the following graph, the original function y = x³ is stretched horizontally by a scale factor of 3 to give the transformed function graph y = (x/3)³. For example, the point (1,1) of the original graph is transformed to (3, 1) of the new graph.

Vertical Dilation

The vertical dilation (also known as vertical scaling) of a function either stretches/shrinks the curve vertically. It changes a function y = f(x) into the form y = k f(x), with a scale factor 'k', parallel to the y-axis. Here,

• If k > 1, then the graph stretches.
• If 0 < k < 1, then the graph shrinks.

In this dilation, there will be changes only in the y-coordinates but there won't be any changes in the x-coordinates. Every old y-coordinate is multiplied by k to find the new y-coordinate. In the following graph, the original function y = x³ is stretched vertically by a scale factor of 3 to give the transformed function graph y = 3x³. For example, the point (1, 1) (on the original graph) corresponds to (1, 3) on the new graph.

A reflection of a function is just the image of the curve with respect to either x-axis or y-axis. This occurs whenever we see the multiplication of a minus sign happening somewhere in the function. Here,

• y = - f(x) is the reflection of y = f(x) with respect to the x-axis.
• y = f(-x) is the reflection of y = f(x) with resepct to the y-axis.

Observe the graph below where the original graph y = (x + 2)² is reflected with respect to each of the x and y axes.

Here, note that when the function is reflected

• with respect to the x-axis, only the signs of the y-coordinates are changed and there is no change in x-coordinates.
• with respect to the y-axis, only the signs of the x-coordinates are changed and there is no change in y-coordinates.

So far we have understood the types of transformations of functions and how do addition/subtraction/multiplication/division of a number and the multiplication of a minus sign would reflect a graph. Let us tabulate all function transformation rules together.

Are the above rules are confusing and difficult to remember? Let us see some important tips to remember these rules.

Tips and Tricks to Remember Function Transformations:

• If some operation is inside the bracket, note that it is related to "horizontal" and in this case, things would happen reverse than what we think.
  For example, we may think f(x + 2) transforms f(x) to the right because it is + but it actually moves left by 2 units.
  In the same way, we may think f(3x) stretches f(x) but no, it shrinks f(x) by a scale factor of 1/3.
• If some operation is outside the bracket, note that it is related to "vertical" and in this case, things would happen straight (not reverse).
  For example, f(x) + 2 moves f(x) "up" it is a "+" symbol there.
  In the same way, 3 f(x) stretches f(x) by a scale factor of 3 as 3 > 1.
• If some number is being added/subtracted, then its related to "translation". For example, f(x + 2) is a horizontal translation and f(x) + 2 is a vertical translation.
• If some number is being multiplied/divided, then its related to "dilation". For example, f(2x) is a horizontal dilation and 2 f(x) is a vertical dilation.
• Just in case of reflection, it is just the opposite of the first and second tricks here. If the minus sign is inside the bracket, it is with respect to the y-axis and if the minus sign is outside the bracket, it is with respect to the x-axis.

We can use the above rules to describe any function transformation. For example, if the question is what is the effect of transformation g(x) = - 3f(x + 5) + 2 on y = f(x), then first observe the sequence of operations that had to be applied on f(x) to get g(x) and then use the above rules to define the transformations. Here, to get g(x) from f(x)

• first f(x) changes into f(x + 5). i.e., horizontal translation by 5 units to the left.
• Then it changes into 3 f(x + 5). i.e., vertical dilation by a scale factor of 3.
• Then it changes into -3 f(x + 5). i.e., reflection about the x-axis.
• Finally, it changes into -3 f(x + 5) + 2. i.e., vertical translation by 2 units up.

Thus, g(x) is obtained from f(x) by horizontal translation by 5 units to the left, vertical dilation by a scale factor of 3, reflection about the x-axis, and vertical translation by 2 units up. We can describe the transformations of functions by using the above tricks also. Give it a try now.

Identifying the transformation by looking at the original and transformed graphs is easy because just by looking at the graph, we can say that the graph moves up by 2 units or left by 3 units, etc. But when a graph is given, graphing the function transformation is sometimes difficult. The following steps make graphing transformations so easier. Here, we are transforming the function y = f(x) to y = a f(b (x + c)) + d.

• Step 1: Note down some coordinates on the original curve that define its shape. i.e., we now know the old x and y coordinates.
• Step 2: To find the new x-coordinate of each point just set "b (x + c) = old x-coordinate" and solve this for x.
• Step 3: To find the new y-coordinate of each point, just apply all outside operations (of brackets) on the old y-coordinate. i.e., find ay + d to find each new y-coordinate where 'y' is the old y-coordinate.

We can understand these steps better by using the example below.

Example: The following graph represents f(x). Graph the function transformation y = 2 f(x/2) + 3.

Solution:

We can clearly see that (-3, 2), (-1, 2), (2, -1) and (6, 1) are defining the shape of the graph. Let us find the new x and y coordinates of each of these points.

Now, we will plot all old points and new points on the coordinate plane and observe the transformations.

☛ Related Topics:

• Transformation Matrix
• Linear Fractional Transformation

What are Transformations of Functions?

The transformations of functions define how to graph a function is moving and how its shape is being changed. There are basically three types of function transformations: translation, dilation, and reflection.

How Do You Find the Function Transformations?

To find the function transformations we have to identify whether it is a translation, dilation, or reflection or sometimes it is a mixture of some/all the transformations. For a function y = f(x),

• if a number is being added or subtracted inside the bracket then it is a horizontal translation. If the number is negative then the horizontal transformation is happening to the right side. If the number is positive then the horizontal transformation is happening to the left side.
• If a number is being added or subtracted outside the bracket then it is a vertical translation. If the number is positive then the vertical translation is happening toward up. If the number is negative then the vertical translation is happening to the downside.
• If a number is being multiplied or divided inside the brackets then it is horizontal dilation. If the number is > 1, then it is a horizontal shrink. If the number is between 0 and 1, then it is a horizontal stretch.
• If a number is being multiplied or divided outside the brackets then it is vertical dilation. If the number is > 1, then it is a vertical stretch. If the number is between 0 and 1, then it is a vertical shrink.
• If the function is multiplied by the minus sign inside the bracket, then it is a reflection with respect to the y-axis.
• If the function is multiplied by the minus sign outside the bracket, then it is a reflection with respect to the x-axis.

How to Explain the Function Transformations?

To explain the function transformations we have to apply the rules of transformations of functions. For example, 3 f(x + 2) - 5 is obtained by applying the following function transformations on f(x):

• horizontal translation by 2 units left.
• Vertical dilation by a scale factor of 3.
• Vertical translation by 5 units down.

What are the Rules of Transformations of Functions?

The rules of function transformations for each of the translation, dilation, and reflection:

• Horizontal translation: it is of the form f(x + k) and it moves f(x) to k units left if k > 0 and k units right if k < 0.
  Vertical translation: it is of the form f(x) + k and it moves f(x) to k units up if k > 0 and k units down if k < 0.
• Horizontal dilation: It is of the form f(kx) and it shrinks f(x) if k > 1 and stretches f(x) if 0 < k < 1.
  Vertical dilation: It is of the form k f(x) and it shrinks f(x) if 0 < k < 1 and stretches f(x) if k > 1.
• Reflection with respect to the x-axis is of the form - f(x).
  Reflection with respect to the y-axis is of the form f(-x).

What are Different Types of Function Transformations?

There are mainly three types of function transformations.

• Translation: it moves the graph of the original function to either left, right, up, or down.
• Dilation: it either shrinks or stretches the graph of the original function horizontally or vertically.
• Reflection: it reflects the graph of the original function ( in other words it creates the mirror image of the original function) with respect to x or y axes.

What is the Easiest Way of Remembering Function Transformations?

Here is the easiest way of remembering the function transformations. If something is happening inside the bracket then it corresponds to the horizontal transformations. If something is happening outside the brackets then it corresponds to the vertical transformations. If a minus sign is being multiplied either outside or inside the bracket then it corresponds to the reflection.