- Graph functions using vertical and horizontal shifts.
- Graph functions using reflections about the and the
- 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 functionthe functionis shifted verticallyunits. See Figure 2 for an example.
To help you visualize the concept of a vertical shift, consider thatTherefore,is equivalent toEvery unit ofis replaced byso the y-value increases or decreases depending on the value ofThe result is a shift upward or downward.
Given a function a new function where is a constant, is a vertical shift of the function All the output values change by units. If is positive, the graph will shift up. If 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 3 shows the area of open vents(in square feet) throughout the day in hours after midnight,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.
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 in Figure 4, for each input value, the output value has increased by 20, so if we call the new functionwe could write
This notation tells us that, for any value ofcan be found by evaluating the functionat the same input and then adding 20 to the result. This definesas a transformation of the functionin this case a vertical shift up 20 units. Notice that, with a vertical shift, the input values stay the same and only the output values change. See Table 1.
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.
Shifting a Tabular Function Vertically
A functionis given in Table 2. Create a table for the function
The formulatells us that we can find the output values ofby subtracting 3 from the output values ofFor example:
Subtracting 3 from eachvalue, we can complete a table of values foras shown in Table 3.
As with the earlier vertical shift, notice the input values stay the same and only the output values change.
The functiongives the heightof a ball (in meters) thrown upward from the ground afterseconds. Suppose the ball was instead thrown from the top of a 10-m building. Relate this new height functiontoand then find a formula for
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 5.
For example, ifthenis 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
Given a functiona new functionwhereis a constant, is a horizontal shift of the functionIfis positive, the graph will shift right. Ifis negative, the graph will shift left.
Adding a Constant to an Input
Returning to our building airflow example from Figure 3, 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.
We can setto be the original program andto be the revised program.
In the new graph, at each time, the airflow is the same as the original functionwas 2 hours later. For example, in the original functionthe airflow starts to change at 8 a.m., whereas for the functionthe airflow starts to change at 6 a.m. The comparable function values areSee Figure 6. Notice also that the vents first opened toat 10 a.m. under the original plan, while under the new plan the vents reachat
8 a.m., so
In both cases, we see that, becausestarts 2 hours sooner,That means that the same output values are reached when
Note thathas 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 functionuses the same outputs asbut matches those outputs to inputs 2 hours earlier than those ofSaid another way, we must add 2 hours to the input ofto find the corresponding output for
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.
Shifting a Tabular Function Horizontally
A functionis given in Table 4. Create a table for the function
The formulatells us that the output values ofare the same as the output value of when the input value is 3 less than the original value. For example, we know thatTo get the same output from the functionwe will need an input value that is 3 larger. We input a value that is 3 larger forbecause the function takes 3 away before evaluating the function
We continue with the other values to create Table 5.
The result is that the functionhas been shifted to the right by 3. Notice the output values forremain the same as the output values forbut the corresponding input values,have shifted to the right by 3. Specifically, 2 shifted to 5, 4 shifted to 7, 6 shifted to 9, and 8 shifted to 11.
Figure 7 represents both of the functions. We can see the horizontal shift in each point.
Identifying a Horizontal Shift of a Toolkit Function
Figure 8 represents a transformation of the toolkit functionRelate this new functiontoand then find a formula for
Notice that the graph is identical in shape to thefunction, but the x-values are shifted to the right 2 units. The vertex used to be at (0,0), but now the vertex is at (2,0). The graph is the basic quadratic function shifted 2 units to the right, so
Notice how we must input the valueto get the output valuethe x-values must be 2 units larger because of the shift to the right by 2 units. We can then use the definition of thefunction to write a formula forby evaluating
To determine whether the shift isor, consider a single reference point on the graph. For a quadratic, looking at the vertex point is convenient. In the original function,In our shifted function,To obtain the output value of 0 from the functionwe need to decide whether a plus or a minus sign will work to satisfyFor this to work, we will need to subtract 2 units from our input values.
Interpreting Horizontal versus Vertical Shifts
The functiongives the number of gallons of gas required to drivemiles. Interpretand
can be interpreted as adding 10 to the output, gallons. This is the gas required to drivemiles, plus another 10 gallons of gas. The graph would indicate a vertical shift.
can be interpreted as adding 10 to the input, miles. So this is the number of gallons of gas required to drive 10 miles more thanmiles. The graph would indicate a horizontal shift.
Given the functiongraph the original functionand the transformationon 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.
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.
Graphing Combined Vertical and Horizontal Shifts
Givensketch a graph of
The functionis our toolkit absolute value function. We know that this graph has a V shape, with the point at the origin. The graph ofhas transformedin two ways:is a change on the inside of the function, giving a horizontal shift left by 1, and the subtraction by 3 inis a change to the outside of the function, giving a vertical shift down by 3. The transformation of the graph is illustrated in Figure 9.
Let us follow one point of the graph of
- The pointis transformed first by shifting left 1 unit:
- The pointis transformed next by shifting down 3 units:
Figure 10 shows the graph of
Givensketch a graph of
Identifying Combined Vertical and Horizontal Shifts
Write a formula for the graph shown in Figure 11, which is a transformation of the toolkit square root function.
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
Note that this transformation has changed the domain and range of the function. This new graph has domainand range
Write a formula for a transformation of the toolkit reciprocal functionthat shifts the function’s graph one unit to the right and one unit up.
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 12.
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.
Given a function a new functionis a vertical reflection of the functionsometimes called a reflection about (or over, or through) the x-axis.
Given a functiona new functionis a horizontal reflection of the functionsometimes called a reflection about the y-axis.
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 x-axis.
- 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 (a) vertically and (b) horizontally.
Reflecting the graph vertically means that each output value will be reflected over the horizontal t-axis as shown in Figure 13.
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 outputvalues, so the negative sign belongs outside of the function.
Reflecting horizontally means that each input value will be reflected over the vertical axis as shown in Figure 14.
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 domainand rangethe vertical reflection gives thefunction the range and the horizontal reflection gives thefunction the domain
Reflect the graph of (a) vertically and (b) horizontally.
Reflecting a Tabular Function Horizontally and Vertically
A functionis given as Table 6. Create a table for the functions below.
Forthe negative sign outside the function indicates a vertical reflection, so the x-values stay the same and each output value will be the opposite of the original output value. See Table 7.
2 4 6 8 –1 –3 –7 –11
Forthe negative sign inside the function indicates a horizontal reflection, so each input value will be the opposite of the original input value and thevalues stay the same as thevalues. See Table 8.
−2 −4 −6 −8 1 3 7 11
Applying a Learning Model Equation
A common model for learning has an equation similar to where is the percentage of mastery that can be achieved after practice sessions. This is a transformation of the function shown in Figure 15. Sketch a graph of
This equation combines three transformations into one equation.
- A horizontal reflection:
- A vertical reflection:
- A vertical shift:
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 16, 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.
As a model for learning, this function would be limited to a domain ofwith corresponding range
Given the toolkit functiongraphandTake 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 or 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 oforwere reflected over both axes, the result would be the original graph, as shown in Figure 17.
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,is neither even nor odd. Also, the only function that is both even and odd is the constant function
A function is called an even function if for every input
The graph of an even function is symmetric about the axis.
A function is called an odd function if for every input
The graph of an odd function is symmetric about the origin.
Given the formula for a function, determine if the function is even, odd, or neither.
- Determine whether the function satisfiesIf it does, it is even.
- Determine whether the function satisfiesIf it does, it is odd.
- 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 functioneven, odd, or neither?
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.
Becausethis is an odd function.
Consider the graph ofin Figure 18. Notice that the graph is symmetric about the origin. For every pointon the graph, the corresponding pointis also on the graph. For example, (1, 3) is on the graph ofand the corresponding pointis also on the graph.
Is the functioneven, 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 19 shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.
Given a functiona new functionwhereis a constant, is a vertical stretch or vertical compression of the function
- Ifthen the graph will be stretched.
- Ifthen the graph will be compressed.
- Ifthen there will be combination of a vertical stretch or compression with a vertical reflection.
Given a function, graph its vertical stretch.
- Identify the value of
- Multiply all range values by
Ifthe graph is stretched by a factor of
Ifthe graph is compressed by a factor of
Ifthe graph is either stretched or compressed and also reflected about the x-axis.
Graphing a Vertical Stretch
A functionmodels the population of fruit flies. The graph is shown in Figure 20.
A scientist is comparing this population to another population,whose growth follows the same pattern, but is twice as large. Sketch a graph of this population.
Because the population is always twice as large, the new population’s output values are always twice the original function’s output values. Graphically, this is shown in Figure 21.
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 inputthe value of the functionis twice the value of the functionNotice that the effect on the graph is a vertical stretching of the graph, where every point doubles its distance from the horizontal axis. The input values,stay the same while the output values are twice as large as before.
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
- Multiply all of the output values by
Finding a Vertical Compression of a Tabular Function
A functionis given as Table 10. Create a table for the function
The formulatells us that the output values ofare half of the output values ofwith the same inputs. For example, we know thatThen
We do the same for the other values to produce Table 11.
The result is that the functionhas been compressed vertically byEach output value is divided in half, so the graph is half the original height.
Recognizing a Vertical Stretch
The graph in Figure 22 is a transformation of the toolkit functionRelate this new functiontoand then find a formula for
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 thatWith the basic cubic function at the same input, Based on that, it appears that the outputs ofarethe outputs of the functionbecauseFrom this we can fairly safely conclude that
We can write a formula forby using the definition of the function
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.
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.
Given a functionthe formresults in a horizontal stretch or compression. Consider the functionObserve Figure 23. The graph ofis a horizontal stretch of the graph of the functionby a factor of 2. The graph ofis a horizontal compression of the graph of the functionby a factor of 2.
Given a functiona new functionwhereis a constant, is a horizontal stretch or horizontal compression of the function
- Ifthen the graph will be compressed by
- Ifthen the graph will be stretched by
- Ifthen there will be combination of a horizontal stretch or compression with a horizontal reflection.
Given a description of a function, sketch a horizontal compression or stretch.
- Write a formula to represent the function.
- Setwherefor a compression or 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,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.
Symbolically, we could write
See Figure 24 for a graphical comparison of the original population and the compressed population.
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 functionis given as Table 13. Create a table for the function
The formulatells us that the output values forare the same as the output values for the functionat an input half the size. Notice that we do not have enough information to determinebecauseand we do not have a value forin our table. Our input values towill need to be twice as large to get inputs forthat we can evaluate. For example, we can determine
We do the same for the other values to produce Table 14.
Figure 25 shows the graphs of both of these sets of points.
Because each input value has been doubled, the result is that the functionhas been stretched horizontally by a factor of 2.
Recognizing a Horizontal Compression on a Graph
Relate the functiontoin Figure 26.
The graph oflooks like the graph ofhorizontally compressed. Becauseends atandends atwe can see that the values have been compressed bybecauseWe might also notice thatandEither way, we can describe this relationship asThis is a horizontal compression by
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 ofin our function: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.
Write a formula for the toolkit square root function horizontally stretched by a factor of 3.
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 aswhich transformation should we start with? The answer here follows nicely from the order of operations. Given the output value ofwe 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 writefor example, we have to think about how the inputs to the functionrelate to the inputs to the functionSuppose we knowWhat input towould produce that output? In other words, what value ofwill allowWe would needTo solve forwe 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.
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.
When combining vertical transformations written in the formfirst vertically stretch byand then vertically shift by
When combining horizontal transformations written in the formfirst horizontally shift byand then horizontally stretch by
When combining horizontal transformations written in the formfirst horizontally stretch byand then horizontally shift by
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 Table 15 for the functioncreate a table of values for the function
There are three steps to this transformation, and we will work from the inside out. Starting with the horizontal transformations,is a horizontal compression bywhich means we multiply each value bySee Table 16.
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. See Table 17.
Finally, we can apply the vertical shift, which will add 1 to all the output values. See Table 18.
Finding a Triple Transformation of a Graph
Use the graph ofin Figure 27 to sketch a graph of
To simplify, let’s start by factoring out the inside of the function.
By factoring the inside, we can first horizontally stretch by 2, as indicated by theon the inside of the function. Remember that twice the size of 0 is still 0, so the point (0,2) remains at (0,2) while the point (2,0) will stretch to (4,0). See Figure 28.
Next, we horizontally shift left by 2 units, as indicated bySee Figure 29.
Last, we vertically shift down by 3 to complete our sketch, as indicated by theon the outside of the function. See Figure 30.
Access this online resource for additional instruction and practice with transformation of functions.
3.5 Section Exercises
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?
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?
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?
For the following exercises, write a formula for the function obtained when the graph is shifted as described.
is shifted up 1 unit and to the left 2 units.
is shifted down 3 units and to the right 1 unit.
is shifted down 4 units and to the right 3 units.
is shifted up 2 units and to the left 4 units.
For the following exercises, describe how the graph of the function is a transformation of the graph of the original function
For the following exercises, determine the interval(s) on which the function is increasing and decreasing.
For the following exercises, use the graph ofshown in Figure 31 to sketch a graph of each transformation of
For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.
Tabular representations for the functionsandare given below. Writeandas transformations of
Tabular representations for the functionsandare given below. Writeandas transformations of
For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.
For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.
For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.
For the following exercises, determine whether the function is odd, even, or neither.
For the following exercises, describe how the graph of each function is a transformation of the graph of the original function
For the following exercises, write a formula for the functionthat results when the graph of a given toolkit function is transformed as described.
The graph ofis reflected over the-axis and horizontally compressed by a factor of .
The graph ofis reflected over the-axis and horizontally stretched by a factor of 2.
The graph ofis vertically compressed by a factor ofthen shifted to the left 2 units and down 3 units.
The graph ofis vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.
The graph ofis vertically compressed by a factor ofthen shifted to the right 5 units and up 1 unit.
The graph ofis 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.
For the following exercises, use the graph in Figure 32 to sketch the given transformations.