Transformation of Functions
LEARNING OBJECTIVES
By the end of this lesson, you will be able to: Graph functions using vertical and horizontal shifts.
 Graph functions using reflections about the $x$axis and the$y$axis.
 Determine whether a function is even, odd, or neither from its graph.
 Graph functions using compressions and stretches.
 Combine transformations.
We all know that a flat mirror enables us to see an accurate image of ourselves and whatever is behind us. When we tilt the mirror, the images we see may shift horizontally or vertically. But what happens when we bend a flexible mirror? Like a carnival funhouse mirror, it presents us with a distorted image of ourselves, stretched or compressed horizontally or vertically. In a similar way, we can distort or transform mathematical functions to better adapt them to describing objects or processes in the real world. In this section, we will take a look at several kinds of transformations.
Graphing Functions Using Vertical and Horizontal Shifts
Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and equations. One method we can employ is to adapt the basic graphs of the toolkit functions to build new models for a given scenario. There are systematic ways to alter functions to construct appropriate models for the problems we are trying to solve.
Identifying Vertical Shifts
One simple kind of transformation involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a vertical shift, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a functionFigure 2. Vertical shift by
To help you visualize the concept of a vertical shift, consider that
A General Note: Vertical Shift
Given a function
Example 1: Adding a Constant to a Function
To regulate temperature in a green building, airflow vents near the roof open and close throughout the day. Figure 2 shows the area of open ventsSolution
We can sketch a graph of this new function by adding 20 to each of the output values of the original function. This will have the effect of shifting the graph vertically up, as shown in Figure 4.Notice that for each input value, the output value has increased by 20, so if we call the new function
This notation tells us that, for any value of
$t$ 
0  8  10  17  19  24 
$V\left(t\right)$ 
0  0  220  220  0  0 
$S\left(t\right)$ 
20  20  240  240  20  20 
How To: Given a tabular function, create a new row to represent a vertical shift.
 Identify the output row or column.
 Determine the magnitude of the shift.
 Add the shift to the value in each output cell. Add a positive value for up or a negative value for down.
Example 2: Shifting a Tabular Function Vertically
A function
$x$ 
2  4  6  8 
$f\left(x\right)$ 
1  3  7  11 
Solution
The formula
Subtracting 3 from each
$x$ 
2  4  6  8 
$f\left(x\right)$ 
1  3  7  11 
$g\left(x\right)$ 
−2  0  4  8 
The function
Identifying Horizontal Shifts
We just saw that the vertical shift is a change to the output, or outside, of the function. We will now look at how changes to input, on the inside of the function, change its graph and meaning. A shift to the input results in a movement of the graph of the function left or right in what is known as a horizontal shift.Figure 5. Horizontal shift of the function
For example, if
A General Note: Horizontal Shift
Given a function
Example 3: Adding a Constant to an Input
Returning to our building airflow example from Example 2, suppose that in autumn the facilities manager decides that the original venting plan starts too late, and wants to begin the entire venting program 2 hours earlier. Sketch a graph of the new function.
Solution
We can set
In the new graph, at each time, the airflow is the same as the original function
In both cases, we see that, because
Analysis of the Solution
Note that
Horizontal changes or "inside changes" affect the domain of a function (the input) instead of the range and often seem counterintuitive. The new function
How To: Given a tabular function, create a new row to represent a horizontal shift.
 Identify the input row or column.
 Determine the magnitude of the shift.
 Add the shift to the value in each input cell.
Example 4: Shifting a Tabular Function Horizontally
A function
$x$ 
2  4  6  8 
$f\left(x\right)$ 
1  3  7  11 
Solution
The formula
We continue with the other values to create this table.
$x$ 
5  7  9  11 
$x  3$ 
2  4  6  8 
$f\left(x\right)$ 
1  3  7  11 
$g\left(x\right)$ 
1  3  7  11 
The result is that the function
Analysis of the Solution
The graph in Figure 7 represents both of the functions. We can see the horizontal shift in each point.Example 5: Identifying a Horizontal Shift of a Toolkit Function
This graph represents a transformation of the toolkit functionSolution
Notice that the graph is identical in shape to the
Notice how we must input the value
Analysis of the Solution
To determine whether the shift is
Example 6: Interpreting Horizontal versus Vertical Shifts
The function
Solution
Try It 1
Given the function
Graphing Functions Using Reflections about the Axes
Another transformation that can be applied to a function is a reflection over the x or yaxis. A vertical reflection reflects a graph vertically across the xaxis, while a horizontal reflection reflects a graph horizontally across the yaxis. The reflections are shown in Figure 9.
Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the xaxis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the yaxis.
A General Note: Reflections
Given a function
Given a function
How To: Given a function, reflect the graph both vertically and horizontally.
 Multiply all outputs by –1 for a vertical reflection. The new graph is a reflection of the original graph about the xaxis.
 Multiply all inputs by –1 for a horizontal reflection. The new graph is a reflection of the original graph about the yaxis.
Example 7: Reflecting a Graph Horizontally and Vertically
Reflect the graph of
Solution
a. Reflecting the graph vertically means that each output value will be reflected over the horizontal taxis as shown in Figure 10.Because each output value is the opposite of the original output value, we can write
Notice that this is an outside change, or vertical shift, that affects the output
Reflecting horizontally means that each input value will be reflected over the vertical axis as shown in Figure 11.
Because each input value is the opposite of the original input value, we can write
Notice that this is an inside change or horizontal change that affects the input values, so the negative sign is on the inside of the function.
Note that these transformations can affect the domain and range of the functions. While the original square root function has domain
Try It 2
Reflect the graph of
Example 8: Reflecting a Tabular Function Horizontally and Vertically
A function
 $g\left(x\right)=f\left(x\right)$
 $h\left(x\right)=f\left(x\right)$
$x$ 
2  4  6  8 
$f\left(x\right)$ 
1  3  7  11 
Solution

For
$g\left(x\right)$, the negative sign outside the function indicates a vertical reflection, so the xvalues stay the same and each output value will be the opposite of the original output value.$x$2 4 6 8 $g\left(x\right)$–1 –3 –7 –11 
For
$h\left(x\right)$, the negative sign inside the function indicates a horizontal reflection, so each input value will be the opposite of the original input value and the$h\left(x\right)$values stay the same as the$f\left(x\right)$values.$x$−2 −4 −6 −8 $h\left(x\right)$1 3 7 11
Try It 3
$x$ 
−2  0  2  4 
$f\left(x\right)$ 
5  10  15  20 
Using the function
a.
b.
Determining Even and Odd Functions
Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions
We say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an odd function.
Note: A function can be neither even nor odd if it does not exhibit either symmetry. For example,
A General Note: Even and Odd Functions
A function is called an even function if for every input
The graph of an even function is symmetric about the
A function is called an odd function if for every input
The graph of an odd function is symmetric about the origin.
How To: Given the formula for a function, determine if the function is even, odd, or neither.
 Determine whether the function satisfies $f\left(x\right)=f\left(x\right)$. If it does, it is even.
 Determine whether the function satisfies $f\left(x\right)=f\left(x\right)$. If it does, it is odd.
 If the function does not satisfy either rule, it is neither even nor odd.
Example 9: Determining whether a Function Is Even, Odd, or Neither
Is the function
Solution
Without looking at a graph, we can determine whether the function is even or odd by finding formulas for the reflections and determining if they return us to the original function. Let’s begin with the rule for even functions.
This does not return us to the original function, so this function is not even. We can now test the rule for odd functions.
Because
Analysis of the Solution
Consider the graph ofGraphing Functions Using Stretches and Compressions
Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity.
We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. Each change has a specific effect that can be seen graphically.
Vertical Stretches and Compressions
When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a vertical stretch; if the constant is between 0 and 1, we get a vertical compression. The graph below shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.A General Note: Vertical Stretches and Compressions
Given a function
 If $a>1$, then the graph will be stretched.
 If 0 < a < 1, then the graph will be compressed.
 If $a<0$, then there will be combination of a vertical stretch or compression with a vertical reflection.
How To: Given a function, graph its vertical stretch.
 Identify the value of $a$.
 Multiply all range values by $a$.

If
$a>1$, the graph is stretched by a factor of$a$.If
${ 0 }<{ a }<{ 1 }$, the graph is compressed by a factor of$a$.If
$a<0$, the graph is either stretched or compressed and also reflected about the xaxis.
Example 10: Graphing a Vertical Stretch
A function
A scientist is comparing this population to another population,
Solution
Because the population is always twice as large, the new population’s output values are always twice the original function’s output values.
If we choose four reference points, (0, 1), (3, 3), (6, 2) and (7, 0) we will multiply all of the outputs by 2.
The following shows where the new points for the new graph will be located.
Symbolically, the relationship is written as
This means that for any input
How To: Given a tabular function and assuming that the transformation is a vertical stretch or compression, create a table for a vertical compression.
 Determine the value of $a$.
 Multiply all of the output values by $a$.
Example 11: Finding a Vertical Compression of a Tabular Function
A function
$x$ 
2  4  6  8 
$f\left(x\right)$ 
1  3  7  11 
Solution
The formula
We do the same for the other values to produce this table.
$x$ 
$2$ 
$4$ 
$6$ 
$8$ 
$g\left(x\right)$ 
$\frac{1}{2}$ 
$\frac{3}{2}$ 
$\frac{7}{2}$ 
$\frac{11}{2}$ 
Analysis of the Solution
The result is that the function
Try It 5
A function$x$ 
2  4  6  8 
$f\left(x\right)$ 
12  16  20  0 
Example 12: Recognizing a Vertical Stretch
The graph is a transformation of the toolkit function
Solution
When trying to determine a vertical stretch or shift, it is helpful to look for a point on the graph that is relatively clear. In this graph, it appears that
We can write a formula for
Try It 6
Write the formula for the function that we get when we stretch the identity toolkit function by a factor of 3, and then shift it down by 2 units.
SolutionHorizontal Stretches and Compressions
Now we consider changes to the inside of a function. When we multiply a function’s input by a positive constant, we get a function whose graph is stretched or compressed horizontally in relation to the graph of the original function. If the constant is between 0 and 1, we get a horizontal stretch; if the constant is greater than 1, we get a horizontal compression of the function.
Given a function
A General Note: Horizontal Stretches and Compressions
Given a function
 If $b>1$, then the graph will be compressed by$\frac{1}{b}$.
 If $0<b<1$, then the graph will be stretched by$\frac{1}{b}$.
 If $b<0$, then there will be combination of a horizontal stretch or compression with a horizontal reflection.
How To: Given a description of a function, sketch a horizontal compression or stretch.
 Write a formula to represent the function.
 Set $g\left(x\right)=f\left(bx\right)$where$b>1$for a compression or$0<b<1$
for a stretch.
Example 13: Graphing a Horizontal Compression
Suppose a scientist is comparing a population of fruit flies to a population that progresses through its lifespan twice as fast as the original population. In other words, this new population,
Solution
Symbolically, we could write
See below for a graphical comparison of the original population and the compressed population.
Analysis of the Solution
Note that the effect on the graph is a horizontal compression where all input values are half of their original distance from the vertical axis.
Example 14: Finding a Horizontal Stretch for a Tabular Function
A function
$x$ 
2  4  6  8 
$f\left(x\right)$ 
1  3  7  11 
Solution
The formula
We do the same for the other values to produce the table below.
$x$ 
4  8  12  16 
$g\left(x\right)$ 
1  3  7  11 
This figure shows the graphs of both of these sets of points.
Analysis of the Solution
Because each input value has been doubled, the result is that the function
Example 15: Recognizing a Horizontal Compression on a Graph
Relate the functionSolution
The graph of
Analysis of the Solution
Notice that the coefficient needed for a horizontal stretch or compression is the reciprocal of the stretch or compression. So to stretch the graph horizontally by a scale factor of 4, we need a coefficient of
Try It 7
Write a formula for the toolkit square root function horizontally stretched by a factor of 3.
SolutionCombining Vertical and Horizontal Shifts
Now that we have two transformations, we can combine them together. Vertical shifts are outside changes that affect the output (
How To: Given a function and both a vertical and a horizontal shift, sketch the graph.
 Identify the vertical and horizontal shifts from the formula.
 The vertical shift results from a constant added to the output. Move the graph up for a positive constant and down for a negative constant.
 The horizontal shift results from a constant added to the input. Move the graph left for a positive constant and right for a negative constant.
 Apply the shifts to the graph in either order.
Example 16: Graphing Combined Vertical and Horizontal Shifts
Given
The function
Let us follow one point of the graph of
 The point $\left(0,0\right)$is transformed first by shifting left 1 unit:$\left(0,0\right)\to \left(1,0\right)$
 The point $\left(1,0\right)$is transformed next by shifting down 3 units:$\left(1,0\right)\to \left(1,3\right)$
Figure 23 is the graph of
Try It 8
Given
Example 17: Identifying Combined Vertical and Horizontal Shifts
Write a formula for the graph shown in Figure 24, which is a transformation of the toolkit square root function.Solution
The graph of the toolkit function starts at the origin, so this graph has been shifted 1 to the right and up 2. In function notation, we could write that as
Using the formula for the square root function, we can write
Analysis of the Solution
Note that this transformation has changed the domain and range of the function. This new graph has domain
Try It 9
Write a formula for a transformation of the toolkit reciprocal function
Example 18: Applying a Learning Model Equation
A common model for learning has an equation similar toSolution
This equation combines three transformations into one equation.
 A horizontal reflection: $f\left(t\right)={2}^{t}$
 A vertical reflection: $f\left(t\right)={2}^{t}$
 A vertical shift: $f\left(t\right)+1={2}^{t}+1$
We can sketch a graph by applying these transformations one at a time to the original function. Let us follow two points through each of the three transformations. We will choose the points (0, 1) and (1, 2).
 First, we apply a horizontal reflection: (0, 1) (–1, 2).
 Then, we apply a vertical reflection: (0, −1) (1, –2).
 Finally, we apply a vertical shift: (0, 0) (1, 1).
This means that the original points, (0,1) and (1,2) become (0,0) and (1,1) after we apply the transformations.
In Figure 26, the first graph results from a horizontal reflection. The second results from a vertical reflection. The third results from a vertical shift up 1 unit.Analysis of the Solution
As a model for learning, this function would be limited to a domain of
Try It 10
Given the toolkit function
Performing a Sequence of Transformations
When combining transformations, it is very important to consider the order of the transformations. For example, vertically shifting by 3 and then vertically stretching by 2 does not create the same graph as vertically stretching by 2 and then vertically shifting by 3, because when we shift first, both the original function and the shift get stretched, while only the original function gets stretched when we stretch first.
When we see an expression such as
Horizontal transformations are a little trickier to think about. When we write
This format ends up being very difficult to work with, because it is usually much easier to horizontally stretch a graph before shifting. We can work around this by factoring inside the function.
Let’s work through an example.
We can factor out a 2.
Now we can more clearly observe a horizontal shift to the left 2 units and a horizontal compression. Factoring in this way allows us to horizontally stretch first and then shift horizontally.
A General Note: Combining Transformations
When combining vertical transformations written in the form
When combining horizontal transformations written in the form
When combining horizontal transformations written in the form
Horizontal and vertical transformations are independent. It does not matter whether horizontal or vertical transformations are performed first.
Example 19: Finding a Triple Transformation of a Tabular Function
Given the table below for the function
$x$ 
6  12  18  24 
$f\left(x\right)$ 
10  14  15  17 
Solution
There are three steps to this transformation, and we will work from the inside out. Starting with the horizontal transformations,
$x$ 
2  4  6  8 
$f\left(3x\right)$ 
10  14  15  17 
Looking now to the vertical transformations, we start with the vertical stretch, which will multiply the output values by 2. We apply this to the previous transformation.
$x$ 
2  4  6  8 
$2f\left(3x\right)$ 
20  28  30  34 
Finally, we can apply the vertical shift, which will add 1 to all the output values.
$x$ 
2  4  6  8 
$g\left(x\right)=2f\left(3x\right)+1$ 
21  29  31  35 
Example 20: Finding a Triple Transformation of a Graph
Use the graph ofSolution
To simplify, let’s start by factoring out the inside of the function.
Next, we horizontally shift left by 2 units, as indicated by
Last, we vertically shift down by 3 to complete our sketch, as indicated by the
Key Equations
Vertical shift  $g\left(x\right)=f\left(x\right)+k$ (up for $k>0$ ) 
Horizontal shift  $g\left(x\right)=f\left(xh\right)$ (right for $h>0$ ) 
Vertical reflection  $g\left(x\right)=f\left(x\right)$ 
Horizontal reflection  $g\left(x\right)=f\left(x\right)$ 
Vertical stretch  $g\left(x\right)=af\left(x\right)$ ( $a>0$ ) 
Vertical compression  $g\left(x\right)=af\left(x\right)$ $\left(0<a<1\right)$ 
Horizontal stretch  $g\left(x\right)=f\left(bx\right)$ $\left(0<b<1\right)$ 
Horizontal compression  $g\left(x\right)=f\left(bx\right)$ ( $b>1$ ) 
Key Concepts
 A function can be shifted vertically by adding a constant to the output.
 A function can be shifted horizontally by adding a constant to the input.
 Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts.
 Vertical and horizontal shifts are often combined.
 A vertical reflection reflects a graph about the $x\text{}$axis. A graph can be reflected vertically by multiplying the output by –1.
 A horizontal reflection reflects a graph about the $y\text{}$axis. A graph can be reflected horizontally by multiplying the input by –1.
 A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph.
 A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly.
 A function presented as an equation can be reflected by applying transformations one at a time.
 Even functions are symmetric about the $y\text{}$axis, whereas odd functions are symmetric about the origin.
 Even functions satisfy the condition $f\left(x\right)=f\left(x\right)$.
 Odd functions satisfy the condition $f\left(x\right)=f\left(x\right)$.
 A function can be odd, even, or neither.
 A function can be compressed or stretched vertically by multiplying the output by a constant.
 A function can be compressed or stretched horizontally by multiplying the input by a constant.
 The order in which different transformations are applied does affect the final function. Both vertical and horizontal transformations must be applied in the order given. However, a vertical transformation may be combined with a horizontal transformation in any order.
Glossary
 even function
 a function whose graph is unchanged by horizontal reflection, $f\left(x\right)=f\left(x\right)$, and is symmetric about the$y\text{}$axis
 horizontal compression
 a transformation that compresses a function’s graph horizontally, by multiplying the input by a constant $b>1$
 horizontal reflection
 a transformation that reflects a function’s graph across the yaxis by multiplying the input by $1$
 horizontal shift
 a transformation that shifts a function’s graph left or right by adding a positive or negative constant to the input
 horizontal stretch
 a transformation that stretches a function’s graph horizontally by multiplying the input by a constant $0<b<1$
 odd function
 a function whose graph is unchanged by combined horizontal and vertical reflection, $f\left(x\right)=f\left(x\right)$, and is symmetric about the origin
 vertical compression
 a function transformation that compresses the function’s graph vertically by multiplying the output by a constant $0<a<1$
 vertical reflection
 a transformation that reflects a function’s graph across the xaxis by multiplying the output by $1$
 vertical shift
 a transformation that shifts a function’s graph up or down by adding a positive or negative constant to the output
 vertical stretch
 a transformation that stretches a function’s graph vertically by multiplying the output by a constant $a>1$
1. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal shift from a vertical shift?
2. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal stretch from a vertical stretch?
3. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal compression from a vertical compression?
4. When examining the formula of a function that is the result of multiple transformations, how can you tell a reflection with respect to the xaxis from a reflection with respect to the yaxis?
5. How can you determine whether a function is odd or even from the formula of the function?
6. Write a formula for the function obtained when the graph of
7. Write a formula for the function obtained when the graph of
is shifted down 3 units and to the right 1 unit.
8. Write a formula for the function obtained when the graph of
9. Write a formula for the function obtained when the graph of
For the following exercises, describe how the graph of the function is a transformation of the graph of the original function
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For the following exercises, determine the interval(s) on which the function is increasing and decreasing.
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For the following exercises, use the graph of
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For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.
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31. Tabular representations for the functions
$x$ 
−2  −1  0  1  2 
$f\left(x\right)$ 
−2  −1  −3  1  2 
$x$ 
−1  0  1  2  3 
$g\left(x\right)$ 
−2  −1  −3  1  2 
$x$ 
−2  −1  0  1  2 
$h\left(x\right)$ 
−1  0  −2  2  3 
32. Tabular representations for the functions
$x$ 
−2  −1  0  1  2 
$f\left(x\right)$ 
−1  −3  4  2  1 
$x$ 
−3  −2  −1  0  1 
$g\left(x\right)$ 
−1  −3  4  2  1 
$x$ 
−2  −1  0  1  2 
$h\left(x\right)$ 
−2  −4  3  1  0 
For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.
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For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.
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For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.
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For the following exercises, determine whether the function is odd, even, or neither.
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For the following exercises, describe how the graph of each function is a transformation of the graph of the original function
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For the following exercises, write a formula for the function
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For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.
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For the following exercises, use the graph below to sketch the given transformations.
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Analysis of the Solution
As with the earlier vertical shift, notice the input values stay the same and only the output values change.