Rewrite Y Squareroot 4x 16 5 to Make It Easy to Graph Using a Translation Describe the Graph

Learning Objectives

In this section, you will:

• Graph functions using vertical and horizontal shifts.
• Graph functions using reflections about the[latex]\,x\text{-axis}\,[/latex] and the[latex]\,y\text{-axis}.[/latex]
• Determine whether a function is even, odd, or neither from its graph.
• Graph functions using compressions and stretches.
• Combine transformations.

Figure 1. (credit: "Misko"/Flickr)

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 function[latex]\,g\left(x\right)=f\left(x\right)+k,\,[/latex]the function[latex]\,f\left(x\right)\,[/latex]is shifted vertically[latex]\,k\,[/latex]units. See (Figure) for an example.

Figure 2. Vertical shift by[latex]\,k=1\,[/latex]of the cube root function[latex]\,f\left(x\right)=\sqrt[3]{x}.[/latex]

To help you visualize the concept of a vertical shift, consider that[latex]\,y=f\left(x\right).\,[/latex]Therefore,[latex]\,f\left(x\right)+k\,[/latex]is equivalent to[latex]\,y+k.\,[/latex]Every unit of[latex]\,y\,[/latex]is replaced by[latex]\,y+k,\,[/latex]so the y-value increases or decreases depending on the value of[latex]\,k.\,[/latex]The result is a shift upward or downward.

Vertical Shift

Given a function [latex]f\left(x\right),[/latex] a new function [latex]g\left(x\right)=f\left(x\right)+k,[/latex] where [latex]\,k[/latex] is a constant, is a vertical shift of the function [latex]f\left(x\right).[/latex] All the output values change by [latex]k[/latex] units. If [latex]k[/latex] is positive, the graph will shift up. If [latex]k[/latex] is negative, the graph will shift down.

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) shows the area of open vents[latex]\,V\,[/latex](in square feet) throughout the day in hours after midnight,[latex]\,t.\,[/latex]During the summer, the facilities manager decides to try to better regulate temperature by increasing the amount of open vents by 20 square feet throughout the day and night. Sketch a graph of this new function.

Figure 3.

How To

Given a tabular function, create a new row to represent a vertical shift.

1. Identify the output row or column.
2. Determine the magnitude of the shift.
3. Add the shift to the value in each output cell. Add a positive value for up or a negative value for down.

Shifting a Tabular Function Vertically

A function[latex]\,f\left(x\right)\,[/latex]is given in (Figure). Create a table for the function[latex]\,g\left(x\right)=f\left(x\right)-3.[/latex]

[latex]x[/latex]	2	4	6	8
[latex]f\left(x\right)[/latex]	1	3	7	11

Analysis

As with the earlier vertical shift, notice the input values stay the same and only the output values change.

Try It

The function[latex]\,h\left(t\right)=-4.9{t}^{2}+30t\,[/latex]gives the height[latex]\,h\,[/latex]of a ball (in meters) thrown upward from the ground after[latex]\,t\,[/latex]seconds. Suppose the ball was instead thrown from the top of a 10-m building. Relate this new height function[latex]\,b\left(t\right)\,[/latex]to[latex]\,h\left(t\right),\,[/latex]and then find a formula for[latex]\,b\left(t\right).[/latex]

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, shown in (Figure).

Figure 5. Horizontal shift of the function[latex]\,f\left(x\right)=\sqrt[3]{x}.\,[/latex]Note that[latex]\,h=+1\,[/latex]shifts the graph to the left, that is, towards negative values of[latex]\,x.[/latex]

For example, if[latex]\,f\left(x\right)={x}^{2},\,[/latex]then[latex]\,g\left(x\right)={\left(x-2\right)}^{2}\,[/latex]is a new function. Each input is reduced by 2 prior to squaring the function. The result is that the graph is shifted 2 units to the right, because we would need to increase the prior input by 2 units to yield the same output value as given in[latex]\,f.[/latex]

Horizontal Shift

Given a function[latex]\,f,\,[/latex]a new function[latex]\,g\left(x\right)=f\left(x-h\right),\,[/latex]where[latex]\,h\,[/latex]is a constant, is a horizontal shift of the function[latex]\,f.\,[/latex]If[latex]\,h\,[/latex]is positive, the graph will shift right. If[latex]\,h\,[/latex]is negative, the graph will shift left.

Adding a Constant to an Input

Returning to our building airflow example from (Figure), 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.

Analysis

Note that[latex]\,V\left(t+2\right)\,[/latex]has the effect of shifting the graph to the left.

Horizontal changes or "inside changes" affect the domain of a function (the input) instead of the range and often seem counterintuitive. The new function[latex]\,F\left(t\right)\,[/latex]uses the same outputs as[latex]\,V\left(t\right),\,[/latex]but matches those outputs to inputs 2 hours earlier than those of[latex]\,V\left(t\right).\,[/latex]Said another way, we must add 2 hours to the input of[latex]\,V\,[/latex]to find the corresponding output for[latex]F:F\left(t\right)=V\left(t+2\right).[/latex]

How To

Given a tabular function, create a new row to represent a horizontal shift.

1. Identify the input row or column.
2. Determine the magnitude of the shift.
3. Add the shift to the value in each input cell.

Shifting a Tabular Function Horizontally

A function[latex]\,f\left(x\right)\,[/latex]is given in (Figure). Create a table for the function[latex]\,g\left(x\right)=f\left(x-3\right).[/latex]

[latex]x[/latex]	2	4	6	8
[latex]f\left(x\right)[/latex]	1	3	7	11

Analysis

(Figure) represents both of the functions. We can see the horizontal shift in each point.

Figure 7.

Identifying a Horizontal Shift of a Toolkit Function

(Figure) represents a transformation of the toolkit function[latex]\,f\left(x\right)={x}^{2}.\,[/latex]Relate this new function[latex]\,g\left(x\right)\,[/latex]to[latex]\,f\left(x\right),\,[/latex]and then find a formula for[latex]\,g\left(x\right).[/latex]

Figure 8.

Analysis

To determine whether the shift is[latex]\,+2\,[/latex]or[latex]\,-2[/latex], consider a single reference point on the graph. For a quadratic, looking at the vertex point is convenient. In the original function,[latex]\,f\left(0\right)=0.\,[/latex]In our shifted function,[latex]\,g\left(2\right)=0.\,[/latex]To obtain the output value of 0 from the function[latex]\,f,\,[/latex]we need to decide whether a plus or a minus sign will work to satisfy[latex]\,g\left(2\right)=f\left(x-2\right)=f\left(0\right)=0.\,[/latex]For this to work, we will need to subtract 2 units from our input values.

Interpreting Horizontal versus Vertical Shifts

The function[latex]\,G\left(m\right)\,[/latex]gives the number of gallons of gas required to drive[latex]\,m\,[/latex]miles. Interpret[latex]\,G\left(m\right)+10\,[/latex]and[latex]\,G\left(m+10\right).[/latex]

Try It

Given the function[latex]\,f\left(x\right)=\sqrt{x},\,[/latex]graph the original function[latex]\,f\left(x\right)\,[/latex]and the transformation[latex]\,g\left(x\right)=f\left(x+2\right)\,[/latex]on the same axes. Is this a horizontal or a vertical shift? Which way is the graph shifted and by how many units?

Combining Vertical and Horizontal Shifts

Now that we have two transformations, we can combine them. Vertical shifts are outside changes that affect the output (y-) values and shift the function up or down. Horizontal shifts are inside changes that affect the input (x-) values and shift the function left or right. Combining the two types of shifts will cause the graph of a function to shift up or down and left or right.

How To

Given a function and both a vertical and a horizontal shift, sketch the graph.

1. Identify the vertical and horizontal shifts from the formula.
2. 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.
3. 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.
4. Apply the shifts to the graph in either order.

Graphing Combined Vertical and Horizontal Shifts

Given[latex]\,f\left(x\right)=|x|,\,[/latex]sketch a graph of[latex]\,h\left(x\right)=f\left(x+1\right)-3.[/latex]

Try It

Given[latex]\,f\left(x\right)=|x|,\,[/latex]sketch a graph of[latex]\,h\left(x\right)=f\left(x-2\right)+4.[/latex]

Show Solution

Identifying Combined Vertical and Horizontal Shifts

Write a formula for the graph shown in (Figure), which is a transformation of the toolkit square root function.

Figure 11.

Analysis

Note that this transformation has changed the domain and range of the function. This new graph has domain[latex]\,\left[1,\infty \right)\,[/latex]and range[latex]\,\left[2,\infty \right).[/latex]

Try It

Write a formula for a transformation of the toolkit reciprocal function[latex]\,f\left(x\right)=\frac{1}{x}\,[/latex]that shifts the function's graph one unit to the right and one unit up.

Show Solution

[latex]g\left(x\right)=\frac{1}{x-1}+1[/latex]

Graphing Functions Using Reflections about the Axes

Another transformation that can be applied to a function is a reflection over the x– or y-axis. A vertical reflection reflects a graph vertically across the x-axis, while a horizontal reflection reflects a graph horizontally across the y-axis. The reflections are shown in (Figure).

Figure 12. Vertical and horizontal reflections of a function.

Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the x-axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the y-axis.

Reflections

Given a function[latex]\,f\left(x\right),[/latex] a new function[latex]\,g\left(x\right)=-f\left(x\right)\,[/latex]is a vertical reflection of the function[latex]\,f\left(x\right),\,[/latex]sometimes called a reflection about (or over, or through) the x-axis.

Given a function[latex]\,f\left(x\right),\,[/latex]a new function[latex]\,g\left(x\right)=f\left(-x\right)\,[/latex]is a horizontal reflection of the function[latex]\,f\left(x\right),\,[/latex]sometimes called a reflection about the y-axis.

How To

Given a function, reflect the graph both vertically and horizontally.

1. Multiply all outputs by –1 for a vertical reflection. The new graph is a reflection of the original graph about the x-axis.
2. Multiply all inputs by –1 for a horizontal reflection. The new graph is a reflection of the original graph about the y-axis.

Reflecting a Graph Horizontally and Vertically

Reflect the graph of[latex]\,s\left(t\right)=\sqrt{t}[/latex] (a) vertically and (b) horizontally.

Try It

Reflect the graph of[latex]\,f\left(x\right)=|x-1|[/latex] (a) vertically and (b) horizontally.

Reflecting a Tabular Function Horizontally and Vertically

A function[latex]\,f\left(x\right)\,[/latex]is given as (Figure). Create a table for the functions below.

1. [latex]\,g\left(x\right)=-f\left(x\right)[/latex]
2. [latex]\,h\left(x\right)=f\left(-x\right)[/latex]

[latex]x[/latex]	2	4	6	8
[latex]f\left(x\right)[/latex]	1	3	7	11

Try It

A function[latex]\,f\left(x\right)\,[/latex]is given as (Figure). Create a table for the functions below.

1. [latex]g\left(x\right)=-f\left(x\right)[/latex]
2. [latex]h\left(x\right)=f\left(-x\right)[/latex]

[latex]x[/latex]	−2	0	2	4
[latex]f\left(x\right)[/latex]	5	10	15	20

Applying a Learning Model Equation

A common model for learning has an equation similar to [latex]k\left(t\right)=-{2}^{-t}+1,\,[/latex] where [latex]k[/latex] is the percentage of mastery that can be achieved after [latex]t[/latex] practice sessions. This is a transformation of the function [latex]f\left(t\right)={2}^{t}[/latex] shown in (Figure). Sketch a graph of [latex]k\left(t\right).[/latex]

Figure 15.

Analysis

As a model for learning, this function would be limited to a domain of[latex]\,t\ge 0,\,[/latex]with corresponding range[latex]\,\left[0,1\right).[/latex]

Try It

Given the toolkit function[latex]\,f\left(x\right)={x}^{2},\,[/latex]graph[latex]\,g\left(x\right)=-f\left(x\right)\,[/latex]and[latex]\,h\left(x\right)=f\left(-x\right).\,[/latex]Take note of any surprising behavior for these functions.

Determining Even and Odd Functions

Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions [latex]f\left(x\right)={x}^{2}[/latex] or [latex]f\left(x\right)=|x|[/latex] will result in the original graph. We say that these types of graphs are symmetric about the y-axis. A function whose graph is symmetric about the y-axis is called an even function.

If the graphs of[latex]\,f\left(x\right)={x}^{3}\,[/latex]or[latex]\,f\left(x\right)=\frac{1}{x}\,[/latex]were reflected over both axes, the result would be the original graph, as shown in (Figure).

Figure 17. (a) The cubic toolkit function (b) Horizontal reflection of the cubic toolkit function (c) Horizontal and vertical reflections reproduce the original cubic function.

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,[latex]\,f\left(x\right)={2}^{x}\,[/latex]is neither even nor odd. Also, the only function that is both even and odd is the constant function[latex]\,f\left(x\right)=0.[/latex]

Even and Odd Functions

A function is called an even function if for every input[latex]\,x[/latex]

[latex]f\left(x\right)=f\left(-x\right)[/latex]

The graph of an even function is symmetric about the [latex]y\text{-}[/latex]axis.

A function is called an odd function if for every input[latex]\,x[/latex]

[latex]f\left(x\right)=-f\left(-x\right)[/latex]

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.

1. Determine whether the function satisfies[latex]\,f\left(x\right)=f\left(-x\right).\,[/latex]If it does, it is even.
2. Determine whether the function satisfies[latex]\,f\left(x\right)=-f\left(-x\right).\,[/latex]If it does, it is odd.
3. If the function does not satisfy either rule, it is neither even nor odd.

Determining whether a Function Is Even, Odd, or Neither

Is the function[latex]\,f\left(x\right)={x}^{3}+2x\,[/latex]even, odd, or neither?

Analysis

Consider the graph of[latex]\,f\,[/latex]in (Figure). Notice that the graph is symmetric about the origin. For every point[latex]\,\left(x,y\right)\,[/latex]on the graph, the corresponding point[latex]\,\left(-x,-y\right)\,[/latex]is also on the graph. For example, (1, 3) is on the graph of[latex]\,f,\,[/latex]and the corresponding point[latex]\left(-1,-3\right)[/latex]is also on the graph.

Figure 18.

Try It

Is the function[latex]\,f\left(s\right)={s}^{4}+3{s}^{2}+7\,[/latex]even, odd, or neither?

Graphing 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. (Figure) shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.

Figure 19. Vertical stretch and compression

Vertical Stretches and Compressions

Given a function[latex]\,f\left(x\right),\,[/latex]a new function[latex]\,g\left(x\right)=af\left(x\right),\,[/latex]where[latex]\,a\,[/latex]is a constant, is a vertical stretch or vertical compression of the function[latex]\,f\left(x\right).[/latex]

• If[latex]\,a>1,\,[/latex]then the graph will be stretched.
• If[latex]\,0<a<1,\,[/latex]then the graph will be compressed.
• If[latex]\,a<0,\,[/latex]then there will be combination of a vertical stretch or compression with a vertical reflection.

How To

Given a function, graph its vertical stretch.

1. Identify the value of[latex]\,a.[/latex]
2. Multiply all range values by[latex]\,a.[/latex]

3. If[latex]\,\,a>1,\,\,[/latex]the graph is stretched by a factor of[latex]\,a.[/latex]

   If[latex]\,0<a<1,\,[/latex]the graph is compressed by a factor of[latex]\,a.[/latex]

   If[latex]\,a<0,\,[/latex]the graph is either stretched or compressed and also reflected about the x-axis.

Graphing a Vertical Stretch

A function[latex]\,P\left(t\right)\,[/latex]models the population of fruit flies. The graph is shown in (Figure).

Figure 20.

A scientist is comparing this population to another population,[latex]\,Q,\,[/latex]whose growth follows the same pattern, but is twice as large. Sketch a graph of this population.

How To

Given a tabular function and assuming that the transformation is a vertical stretch or compression, create a table for a vertical compression.

1. Determine the value of [latex]\,a.[/latex]
2. Multiply all of the output values by[latex]\,a.[/latex]

Finding a Vertical Compression of a Tabular Function

A function[latex]\,f\,[/latex]is given as (Figure). Create a table for the function[latex]\,g\left(x\right)=\frac{1}{2}f\left(x\right).[/latex]

[latex]x[/latex]	2	4	6	8
[latex]f\left(x\right)\,[/latex]	1	3	7	11

Analysis

The result is that the function[latex]\,g\left(x\right)\,[/latex]has been compressed vertically by[latex]\,\frac{1}{2}.\,[/latex]Each output value is divided in half, so the graph is half the original height.

Try It

A function[latex]\,f\,[/latex]is given as (Figure). Create a table for the function[latex]\,g\left(x\right)=\frac{3}{4}f\left(x\right).[/latex]

[latex]x[/latex]	2	4	6	8
[latex]f\left(x\right)[/latex]	12	16	20	0

Recognizing a Vertical Stretch

The graph in (Figure) is a transformation of the toolkit function[latex]\,f\left(x\right)={x}^{3}.\,[/latex]Relate this new function[latex]\,g\left(x\right)\,[/latex]to[latex]\,f\left(x\right),\,[/latex]and then find a formula for[latex]\,g\left(x\right).[/latex]

Figure 22.

Try It

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.

Show Solution

[latex]g\left(x\right)=3x-2[/latex]

Horizontal 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.

Figure 23.

Given a function[latex]\,y=f\left(x\right),\,[/latex]the form[latex]\,y=f\left(bx\right)\,[/latex]results in a horizontal stretch or compression. Consider the function[latex]\,y={x}^{2}.\,[/latex]Observe (Figure). The graph of[latex]\,y={\left(0.5x\right)}^{2}\,[/latex]is a horizontal stretch of the graph of the function[latex]\,y={x}^{2}\,[/latex]by a factor of 2. The graph of[latex]\,y={\left(2x\right)}^{2}\,[/latex]is a horizontal compression of the graph of the function[latex]\,y={x}^{2}\,[/latex]by a factor of 2.

Horizontal Stretches and Compressions

Given a function[latex]\,f\left(x\right),\,[/latex]a new function[latex]\,g\left(x\right)=f\left(bx\right),\,[/latex]where[latex]\,b\,[/latex]is a constant, is a horizontal stretch or horizontal compression of the function[latex]\,f\left(x\right).[/latex]

• If[latex]\,b>1,\,[/latex]then the graph will be compressed by[latex]\,\frac{1}{b}.[/latex]
• If[latex]\,0<b<1,\,[/latex]then the graph will be stretched by[latex]\,\frac{1}{b}.[/latex]
• If[latex]\,b<0,\,[/latex]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.

1. Write a formula to represent the function.
2. Set[latex]\,g\left(x\right)=f\left(bx\right)\,[/latex]where[latex]\,b>1\,[/latex]for a compression or[latex]\,0<b<1\,[/latex]
   for a stretch.

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,[latex]\,R,\,[/latex]will progress in 1 hour the same amount as the original population does in 2 hours, and in 2 hours, it will progress as much as the original population does in 4 hours. Sketch a graph of this population.

Analysis

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.

Finding a Horizontal Stretch for a Tabular Function

A function[latex]\,f\left(x\right)\,[/latex]is given as (Figure). Create a table for the function[latex]\,g\left(x\right)=f\left(\frac{1}{2}x\right).[/latex]

[latex]x[/latex]	2	4	6	8
[latex]f\left(x\right)[/latex]	1	3	7	11

Analysis

Because each input value has been doubled, the result is that the function[latex]\,g\left(x\right)\,[/latex]has been stretched horizontally by a factor of 2.

Recognizing a Horizontal Compression on a Graph

Relate the function[latex]\,g\left(x\right)\,[/latex]to[latex]\,f\left(x\right)\,[/latex]in (Figure).

Figure 26.

Show Solution

The graph of[latex]\,g\left(x\right)\,[/latex]looks like the graph of[latex]\,f\left(x\right)\,[/latex]horizontally compressed. Because[latex]\,f\left(x\right)\,[/latex]ends at[latex]\,\left(6,4\right)\,[/latex]and[latex]\,g\left(x\right)\,[/latex]ends at[latex]\,\left(2,4\right),\,[/latex]we can see that the [latex]\,x\text{-}[/latex]values have been compressed by[latex]\,\frac{1}{3},\,[/latex]because[latex]\,6\left(\frac{1}{3}\right)=2.\,[/latex]We might also notice that[latex]\,g\left(2\right)=f\left(6\right)\,[/latex]and[latex]\,g\left(1\right)=f\left(3\right).\,[/latex]Either way, we can describe this relationship as[latex]\,g\left(x\right)=f\left(3x\right).\,[/latex]This is a horizontal compression by[latex]\,\frac{1}{3}.[/latex]

Analysis

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[latex]\,\frac{1}{4}\,[/latex]in our function:[latex]\,f\left(\frac{1}{4}x\right).\,[/latex]This means that the input values must be four times larger to produce the same result, requiring the input to be larger, causing the horizontal stretching.

Try It

Write a formula for the toolkit square root function horizontally stretched by a factor of 3.

Show Solution

[latex]g\left(x\right)=f\left(\frac{1}{3}x\right)\,[/latex]so using the square root function we get[latex]\,g\left(x\right)=\sqrt{\frac{1}{3}x}[/latex]

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[latex]\,\,2f\left(x\right)+3,\,[/latex]which transformation should we start with? The answer here follows nicely from the order of operations. Given the output value of[latex]\,f\left(x\right),\,[/latex]we first multiply by 2, causing the vertical stretch, and then add 3, causing the vertical shift. In other words, multiplication before addition.

Horizontal transformations are a little trickier to think about. When we write[latex]\,g\left(x\right)=f\left(2x+3\right),\,[/latex]for example, we have to think about how the inputs to the function[latex]\,g\,[/latex]relate to the inputs to the function[latex]\,\,f.\,[/latex]Suppose we know[latex]\,f\left(7\right)=12.\,[/latex]What input to[latex]\,g\,[/latex]would produce that output? In other words, what value of[latex]\,x\,[/latex]will allow[latex]\,g\left(x\right)=f\left(2x+3\right)=12?\,[/latex]We would need[latex]\,2x+3=7.\,[/latex]To solve for[latex]\,x,\,[/latex]we would first subtract 3, resulting in a horizontal shift, and then divide by 2, causing a horizontal compression.

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.

[latex]f\left(bx+p\right)=f\left(b\left(x+\frac{p}{b}\right)\right)[/latex]

Let's work through an example.

[latex]f\left(x\right)={\left(2x+4\right)}^{2}[/latex]

We can factor out a 2.

[latex]f\left(x\right)={\left(2\left(x+2\right)\right)}^{2}[/latex]

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.

Combining Transformations

When combining vertical transformations written in the form[latex]\,af\left(x\right)+k,\,[/latex]first vertically stretch by[latex]\,a\,[/latex]and then vertically shift by[latex]\,k.[/latex]

When combining horizontal transformations written in the form[latex]\,f\left(bx-h\right),\,[/latex]first horizontally shift by[latex]\,h\,[/latex]and then horizontally stretch by[latex]\,\frac{1}{b}.[/latex]

When combining horizontal transformations written in the form[latex]\,f\left(b\left(x-h\right)\right),\,[/latex]first horizontally stretch by[latex]\,\frac{1}{b}\,[/latex]and then horizontally shift by[latex]\,h.[/latex]

Horizontal and vertical transformations are independent. It does not matter whether horizontal or vertical transformations are performed first.

Finding a Triple Transformation of a Tabular Function

Given (Figure) for the function[latex]\,f\left(x\right),\,[/latex]create a table of values for the function[latex]\,g\left(x\right)=2f\left(3x\right)+1.[/latex]

[latex]x[/latex]	6	12	18	24
[latex]f\left(x\right)[/latex]	10	14	15	17

Finding a Triple Transformation of a Graph

Use the graph of[latex]\,f\left(x\right)\,[/latex]in (Figure) to sketch a graph of[latex]\,k\left(x\right)=f\left(\frac{1}{2}x+1\right)-3.[/latex]

Figure 27.

Key Equations

Vertical shift	[latex]g\left(x\right)=f\left(x\right)+k\,[/latex](up for[latex]\,k>0[/latex])
Horizontal shift	[latex]g\left(x\right)=f\left(x-h\right)[/latex](right for[latex]\,h>0[/latex])
Vertical reflection	[latex]g\left(x\right)=-f\left(x\right)[/latex]
Horizontal reflection	[latex]g\left(x\right)=f\left(-x\right)[/latex]
Vertical stretch	[latex]g\left(x\right)=af\left(x\right)\,[/latex]([latex]a>0[/latex] )
Vertical compression	[latex]g\left(x\right)=af\left(x\right)\,[/latex][latex]\left(0<a<1\right)[/latex]
Horizontal stretch	[latex]g\left(x\right)=f\left(bx\right)[/latex][latex]\left(0<b<1\right)[/latex]
Horizontal compression.	[latex]g\left(x\right)=f\left(bx\right)\,[/latex]([latex]b>1[/latex])

Key Concepts

• A function can be shifted vertically by adding a constant to the output. See (Figure) and (Figure).
• A function can be shifted horizontally by adding a constant to the input. See (Figure), (Figure), and (Figure).
• Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts. See (Figure).
• Vertical and horizontal shifts are often combined. See (Figure) and (Figure).
• A vertical reflection reflects a graph about the [latex]\,x\text{-}[/latex]axis. A graph can be reflected vertically by multiplying the output by –1.
• A horizontal reflection reflects a graph about the [latex]y\text{-}[/latex]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. See (Figure).
• A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly. See (Figure).
• A function presented as an equation can be reflected by applying transformations one at a time. See (Figure).
• Even functions are symmetric about the [latex]y\text{-}[/latex]axis, whereas odd functions are symmetric about the origin.
• Even functions satisfy the condition[latex]\,f\left(x\right)=f\left(-x\right).[/latex]
• Odd functions satisfy the condition[latex]\,f\left(x\right)=-f\left(-x\right).[/latex]
• A function can be odd, even, or neither. See (Figure).
• A function can be compressed or stretched vertically by multiplying the output by a constant. See (Figure), (Figure), and (Figure).
• A function can be compressed or stretched horizontally by multiplying the input by a constant. See (Figure), (Figure), and (Figure).
• 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. See (Figure) and (Figure).

Section Exercises

Verbal

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?

Show Solution

A horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output.

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?

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?

Show Solution

A horizontal compression results when a constant greater than 1 is multiplied by the input. A vertical compression results when a constant between 0 and 1 is multiplied by the output.

When examining the formula of a function that is the result of multiple transformations, how can you tell a reflection with respect to the x-axis from a reflection with respect to the y-axis?

How can you determine whether a function is odd or even from the formula of the function?

Show Solution

For a function[latex]\,f,\,[/latex]substitute[latex]\,\left(-x\right)\,[/latex]for[latex]\,\left(x\right)\,[/latex]in[latex]\,f\left(x\right).\,[/latex]Simplify. If the resulting function is the same as the original function,[latex]\,f\left(-x\right)=f\left(x\right),\,[/latex]then the function is even. If the resulting function is the opposite of the original function,[latex]\,f\left(-x\right)=-f\left(x\right),\,[/latex]then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.

Algebraic

For the following exercises, write a formula for the function obtained when the graph is shifted as described.

[latex]\,f\left(x\right)=\sqrt{x}\,[/latex]is shifted up 1 unit and to the left 2 units.

[latex]\,f\left(x\right)=|x|\,[/latex]
is shifted down 3 units and to the right 1 unit.

Show Solution

[latex]g\left(x\right)=|x-1|-3[/latex]

[latex]\,f\left(x\right)=\frac{1}{x}\,[/latex]is shifted down 4 units and to the right 3 units.

[latex]\,f\left(x\right)=\frac{1}{{x}^{2}}\,[/latex]is shifted up 2 units and to the left 4 units.

Show Solution

[latex]g\left(x\right)=\frac{1}{{\left(x+4\right)}^{2}}+2[/latex]

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function[latex]\,f.[/latex]

[latex]y=f\left(x-49\right)[/latex]

[latex]y=f\left(x+43\right)[/latex]

Show Solution

The graph of[latex]\,f\left(x+43\right)\,[/latex]is a horizontal shift to the left 43 units of the graph of[latex]\,f.[/latex]

[latex]y=f\left(x+3\right)[/latex]

[latex]y=f\left(x-4\right)[/latex]

Show Solution

The graph of[latex]\,f\left(x-4\right)\,[/latex]is a horizontal shift to the right 4 units of the graph of[latex]\,f.[/latex]

[latex]y=f\left(x\right)+5[/latex]

[latex]y=f\left(x\right)+8[/latex]

Show Solution

The graph of[latex]\,f\left(x\right)+8\,[/latex]is a vertical shift up 8 units of the graph of[latex]\,f.[/latex]

[latex]y=f\left(x\right)-2[/latex]

[latex]y=f\left(x\right)-7[/latex]

Show Solution

The graph of[latex]\,f\left(x\right)-7\,[/latex]is a vertical shift down 7 units of the graph of[latex]\,f.[/latex]

[latex]y=f\left(x-2\right)+3[/latex]

[latex]y=f\left(x+4\right)-1[/latex]

Show Solution

The graph of [latex]f\left(x+4\right)-1[/latex] is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of [latex]f.[/latex]

For the following exercises, determine the interval(s) on which the function is increasing and decreasing.

[latex]f\left(x\right)=4{\left(x+1\right)}^{2}-5[/latex]

[latex]g\left(x\right)=5{\left(x+3\right)}^{2}-2[/latex]

Show Solution

decreasing on[latex]\,\left(-\infty ,-3\right)\,[/latex]and increasing on[latex]\,\left(-3,\infty \right)[/latex]

[latex]a\left(x\right)=\sqrt{-x+4}[/latex]

[latex]k\left(x\right)=-3\sqrt{x}-1[/latex]

Show Solution

decreasing on [latex]\left(0,\,\infty \right)[/latex]

Graphical

For the following exercises, use the graph of[latex]\,f\left(x\right)={2}^{x}\,[/latex]shown in (Figure) to sketch a graph of each transformation of[latex]\,f\left(x\right).[/latex]

[latex]g\left(x\right)={2}^{x}+1[/latex]

[latex]h\left(x\right)={2}^{x}-3[/latex]

Show Solution

[latex]w\left(x\right)={2}^{x-1}[/latex]

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.

[latex]f\left(t\right)={\left(t+1\right)}^{2}-3[/latex]

Show Solution

[latex]h\left(x\right)=|x-1|+4[/latex]

[latex]k\left(x\right)={\left(x-2\right)}^{3}-1[/latex]

Show Solution

[latex]m\left(t\right)=3+\sqrt{t+2}[/latex]

Numeric

Tabular representations for the functions[latex]\,f,\,g,\,[/latex]and[latex]\,h\,[/latex]are given below. Write[latex]\,g\left(x\right)\,[/latex]and[latex]\,h\left(x\right)\,[/latex]as transformations of[latex]\,f\left(x\right).[/latex]

[latex]x[/latex]	−2	−1	0	1	2
[latex]f\left(x\right)[/latex]	−2	−1	−3	1	2

[latex]x[/latex]	−1	0	1	2	3
[latex]g\left(x\right)[/latex]	−2	−1	−3	1	2

[latex]x[/latex]	−2	−1	0	1	2
[latex]h\left(x\right)[/latex]	−1	0	−2	2	3

Show Solution

[latex]g\left(x\right)=f\left(x-1\right),\,h\left(x\right)=f\left(x\right)+1[/latex]

Tabular representations for the functions[latex]\,f,\,g,\,[/latex]and[latex]\,h\,[/latex]are given below. Write[latex]\,g\left(x\right)\,[/latex]and[latex]\,h\left(x\right)\,[/latex]as transformations of[latex]\,f\left(x\right).[/latex]

[latex]x[/latex]	−2	−1	0	1	2
[latex]f\left(x\right)[/latex]	−1	−3	4	2	1

[latex]x[/latex]	−3	−2	−1	0	1
[latex]g\left(x\right)[/latex]	−1	−3	4	2	1

[latex]x[/latex]	−2	−1	0	1	2
[latex]h\left(x\right)[/latex]	−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.

Show Solution

[latex]f\left(x\right)=|x-3|-2[/latex]

Show Solution

[latex]f\left(x\right)=\sqrt{x+3}-1[/latex]

Show Solution

[latex]f\left(x\right)={\left(x-2\right)}^{2}[/latex]

Show Solution

[latex]f\left(x\right)=|x+3|-2[/latex]

For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.

Show Solution

[latex]f\left(x\right)=-\sqrt{x}[/latex]

For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.

Show Solution

[latex]f\left(x\right)=-{\left(x+1\right)}^{2}+2[/latex]

Show Solution

[latex]f\left(x\right)=\sqrt{-x}+1[/latex]

For the following exercises, determine whether the function is odd, even, or neither.

[latex]f\left(x\right)=3{x}^{4}[/latex]

[latex]g\left(x\right)=\sqrt{x}[/latex]

[latex]h\left(x\right)=\frac{1}{x}+3x[/latex]

[latex]f\left(x\right)={\left(x-2\right)}^{2}[/latex]

[latex]g\left(x\right)=2{x}^{4}[/latex]

[latex]h\left(x\right)=2x-{x}^{3}[/latex]

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function[latex]\,f.[/latex]

[latex]g\left(x\right)=-f\left(x\right)[/latex]

Show Solution

The graph of[latex]\,g\,[/latex]is a vertical reflection (across the [latex]\,x[/latex]-axis) of the graph of[latex]\,f.[/latex]

[latex]g\left(x\right)=f\left(-x\right)[/latex]

[latex]g\left(x\right)=4f\left(x\right)[/latex]

Show Solution

The graph of[latex]\,g\,[/latex]is a vertical stretch by a factor of 4 of the graph of[latex]\,f.[/latex]

[latex]g\left(x\right)=6f\left(x\right)[/latex]

[latex]g\left(x\right)=f\left(5x\right)[/latex]

Show Solution

The graph of[latex]\,g\,[/latex]is a horizontal compression by a factor of[latex]\,\frac{1}{5}\,[/latex]of the graph of[latex]\,f.[/latex]

[latex]g\left(x\right)=f\left(2x\right)[/latex]

[latex]g\left(x\right)=f\left(\frac{1}{3}x\right)[/latex]

Show Solution

The graph of[latex]\,g\,[/latex]is a horizontal stretch by a factor of 3 of the graph of[latex]\,f.[/latex]

[latex]g\left(x\right)=f\left(\frac{1}{5}x\right)[/latex]

[latex]g\left(x\right)=3f\left(-x\right)[/latex]

Show Solution

The graph of[latex]\,g\,[/latex]is a horizontal reflection across the[latex]\,y[/latex]-axis and a vertical stretch by a factor of 3 of the graph of[latex]\,f.[/latex]

[latex]g\left(x\right)=-f\left(3x\right)[/latex]

For the following exercises, write a formula for the function[latex]\,g\,[/latex]that results when the graph of a given toolkit function is transformed as described.

The graph of[latex]\,f\left(x\right)=|x|\,[/latex]is reflected over the[latex]\,y[/latex]–axis and horizontally compressed by a factor of[latex]\,\frac{1}{4}[/latex]
.

Show Solution

[latex]g\left(x\right)=|-4x|[/latex]

The graph of[latex]\,f\left(x\right)=\sqrt{x}\,[/latex]is reflected over the[latex]\,x[/latex]-axis and horizontally stretched by a factor of 2.

The graph of[latex]\,f\left(x\right)=\frac{1}{{x}^{2}}\,[/latex]is vertically compressed by a factor of[latex]\,\frac{1}{3},\,[/latex]then shifted to the left 2 units and down 3 units.

Show Solution

[latex]g\left(x\right)=\frac{1}{3{\left(x+2\right)}^{2}}-3[/latex]

The graph of[latex]\,f\left(x\right)=\frac{1}{x}\,[/latex]is vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.

The graph of[latex]\,f\left(x\right)={x}^{2}\,[/latex]is vertically compressed by a factor of[latex]\,\frac{1}{2},\,[/latex]then shifted to the right 5 units and up 1 unit.

Show Solution

[latex]g\left(x\right)=\frac{1}{2}{\left(x-5\right)}^{2}+1[/latex]

The graph of[latex]\,f\left(x\right)={x}^{2}\,[/latex]is horizontally stretched by a factor of 3, then shifted to the left 4 units and down 3 units.

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.

[latex]g\left(x\right)=4{\left(x+1\right)}^{2}-5[/latex]

[latex]g\left(x\right)=5{\left(x+3\right)}^{2}-2[/latex]

[latex]h\left(x\right)=-2|x-4|+3[/latex]

[latex]k\left(x\right)=-3\sqrt{x}-1[/latex]

[latex]m\left(x\right)=\frac{1}{2}{x}^{3}[/latex]

[latex]n\left(x\right)=\frac{1}{3}|x-2|[/latex]

[latex]p\left(x\right)={\left(\frac{1}{3}x\right)}^{3}-3[/latex]

[latex]q\left(x\right)={\left(\frac{1}{4}x\right)}^{3}+1[/latex]

[latex]a\left(x\right)=\sqrt{-x+4}[/latex]

For the following exercises, use the graph in (Figure) to sketch the given transformations.

[latex]g\left(x\right)=f\left(x\right)-2[/latex]

[latex]g\left(x\right)=-f\left(x\right)[/latex]

Show Solution

[latex]g\left(x\right)=f\left(x+1\right)[/latex]

[latex]g\left(x\right)=f\left(x-2\right)[/latex]

Show Solution

Glossary

even function

a function whose graph is unchanged by horizontal reflection,[latex]\,f\left(x\right)=f\left(-x\right),\,[/latex]and is symmetric about the [latex]y\text{-}[/latex]axis

horizontal compression

a transformation that compresses a function's graph horizontally, by multiplying the input by a constant[latex]\,b>1[/latex]

horizontal reflection

a transformation that reflects a function's graph across the y-axis by multiplying the input by[latex]\,-1[/latex]

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[latex]\,0<b<1[/latex]

odd function

a function whose graph is unchanged by combined horizontal and vertical reflection,[latex]\,f\left(x\right)=-f\left(-x\right),\,[/latex]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[latex]\,0<a<1[/latex]

vertical reflection

a transformation that reflects a function's graph across the x-axis by multiplying the output by[latex]\,-1[/latex]

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[latex]\,a>1[/latex]

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Source: https://courses.lumenlearning.com/suny-osalgebratrig/chapter/transformation-of-functions/

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