In this section, you will:
-axis and the
-axis.
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.
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.
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
the function
is shifted vertically
units. See [link] for an example.
To help you visualize the concept of a vertical shift, consider that
Therefore,
is equivalent to
Every unit of
is replaced by
so the
value increases or decreases depending on the value of
The 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.
To regulate temperature in a green building, airflow vents near the roof open and close throughout the day. [link] 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 [link].
Notice that in [link], for each input value, the output value has increased by 20, so if we call the new function
we could write
This notation tells us that, for any value of
can be found by evaluating the function
at the same input and then adding 20 to the result. This defines
as a transformation of the function
in 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 [link].
0 | 8 | 10 | 17 | 19 | 24 | |
0 | 0 | 220 | 220 | 0 | 0 | |
20 | 20 | 240 | 240 | 20 | 20 |
Given a tabular function, create a new row to represent a vertical shift.
A function
is given in [link]. Create a table for the function
2 | 4 | 6 | 8 | |
1 | 3 | 7 | 11 |
The formula
tells us that we can find the output values of
by subtracting 3 from the output values of
For example:
Subtracting 3 from each
value, we can complete a table of values for
as shown in [link].
2 | 4 | 6 | 8 | |
1 | 3 | 7 | 11 | |
−2 | 0 | 4 | 8 |
As with the earlier vertical shift, notice the input values stay the same and only the output values change.
The function
gives the height
of a ball (in meters) thrown upward from the ground after
seconds. Suppose the ball was instead thrown from the top of a 10-m building. Relate this new height function
to
and then find a formula for
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 [link].
For example, if
then
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
Given a function
a new function
where
is a constant, is a horizontal shift of the function
If
is positive, the graph will shift right. If
is negative, the graph will shift left.
Returning to our building airflow example from [link], 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 set
to be the original program and
to be the revised program.
In the new graph, at each time, the airflow is the same as the original function
was 2 hours later. For example, in the original function
the airflow starts to change at 8 a.m., whereas for the function
the airflow starts to change at 6 a.m. The comparable function values are
See [link]. Notice also that the vents first opened to
at 10 a.m. under the original plan, while under the new plan the vents reach
at 8 a.m., so
In both cases, we see that, because
starts 2 hours sooner,
That means that the same output values are reached when
Note that
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
uses the same outputs as
but matches those outputs to inputs 2 hours earlier than those of
Said another way, we must add 2 hours to the input of
to find the corresponding output for
Given a tabular function, create a new row to represent a horizontal shift.
A function
is given in [link]. Create a table for the function
2 | 4 | 6 | 8 | |
1 | 3 | 7 | 11 |
The formula
tells us that the output values of
are the same as the output value of
when the input value is 3 less than the original value. For example, we know that
To get the same output from the function
we will need an input value that is 3 larger. We input a value that is 3 larger for
because the function takes 3 away before evaluating the function
We continue with the other values to create [link].
5 | 7 | 9 | 11 | |
2 | 4 | 6 | 8 | |
1 | 3 | 7 | 11 | |
1 | 3 | 7 | 11 |
The result is that the function
has been shifted to the right by 3. Notice the output values for
remain the same as the output values for
but 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.
[link] represents both of the functions. We can see the horizontal shift in each point.
[link] represents a transformation of the toolkit function
Relate this new function
to
and then find a formula for
Notice that the graph is identical in shape to the
function, 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 value
to get the output value
the x-values must be 2 units larger because of the shift to the right by 2 units. We can then use the definition of the
function to write a formula for
by evaluating
To determine whether the shift is
or
, 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 function
we need to decide whether a plus or a minus sign will work to satisfy
For this to work, we will need to subtract 2 units from our input values.
The function
gives the number of gallons of gas required to drive
miles. Interpret
and
can be interpreted as adding 10 to the output, gallons. This is the gas required to drive
miles, 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 than
miles. The graph would indicate a horizontal shift.
Given the function
graph the original function
and the transformation
on the same axes. Is this a horizontal or a vertical shift? Which way is the graph shifted and by how many units?
The graphs of
and
are shown below. The transformation is a horizontal shift. The function is shifted to the left by 2 units.
Now that we have two transformations, we can combine them together. Vertical shifts are outside changes that affect the output (
) axis values and shift the function up or down. Horizontal shifts are inside changes that affect the input (
) axis 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 right or left.
Given a function and both a vertical and a horizontal shift, sketch the graph.
Given
sketch a graph of
The function
is our toolkit absolute value function. We know that this graph has a V shape, with the point at the origin. The graph of
has transformed
in two ways:
is a change on the inside of the function, giving a horizontal shift left by 1, and the subtraction by 3 in
is a change to the outside of the function, giving a vertical shift down by 3. The transformation of the graph is illustrated in [link].
Let us follow one point of the graph of
is transformed first by shifting left 1 unit:
is transformed next by shifting down 3 units:
[link] shows the graph of
Given
sketch a graph of
Write a formula for the graph shown in [link], 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 domain
and range
Write a formula for a transformation of the toolkit reciprocal function
that shifts the function’s graph one unit to the right and one unit up.
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 [link].
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 function
is a vertical reflection of the function
sometimes called a reflection about (or over, or through) the x-axis.
Given a function
a new function
is a horizontal reflection of the function
sometimes called a reflection about the y-axis.
Given a function, reflect the graph both vertically and horizontally.
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 [link].
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
values, 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 [link].
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
and range
the vertical reflection gives the
function the range
and the horizontal reflection gives the
function the domain
Reflect the graph of
(a) vertically and (b) horizontally.
A function
is given as [link]. Create a table for the functions below.
2 | 4 | 6 | 8 | |
1 | 3 | 7 | 11 |
For
the 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 [link].
2 | 4 | 6 | 8 | |
–1 | –3 | –7 | –11 |
For
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
values stay the same as the
values. See [link].
−2 | −4 | −6 | −8 | |
1 | 3 | 7 | 11 |
-2 | 0 | 2 | 4 | |
-2 | 0 | 2 | 4 | |
15 | 10 | 5 | unknown |
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 [link]. Sketch a graph of
This equation combines three transformations into one equation.
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).
This means that the original points, (0,1) and (1,2) become (0,0) and (-1,-1) after we apply the transformations.
In [link], 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 of
with corresponding range
Given the toolkit function
graph
and
Take note of any surprising behavior for these functions.
Notice:
looks the same as
.
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. Functions whose graphs are symmetric about the y-axis are called even functions.
If the graphs of
or
were reflected over both axes, the result would be the original graph, as shown in [link].
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.
If it does, it is even.
If it does, it is odd.
Is the function
even, 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.
Because
this is an odd function.
Consider the graph of
in [link]. Notice that the graph is symmetric about the origin. For every point
on the graph, the corresponding point
is also on the graph. For example, (1, 3) is on the graph of
and the corresponding point
is also on the graph.
Is the function
even, odd, or neither?
even
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.
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. [link] shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.
Given a function
a new function
where
is a constant, is a vertical stretch or vertical compression of the function
then the graph will be stretched.
then the graph will be compressed.
then there will be combination of a vertical stretch or compression with a vertical reflection.
Given a function, graph its vertical stretch.
If
the graph is stretched by a factor of
If
the graph is compressed by a factor of
If
the graph is either stretched or compressed and also reflected about the x-axis.
A function
models the population of fruit flies. The graph is shown in [link].
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 [link].
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
the value of the function
is twice the value of the function
Notice 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.
A function
is given as [link]. Create a table for the function
2 | 4 | 6 | 8 | |
1 | 3 | 7 | 11 |
The formula
tells us that the output values of
are half of the output values of
with the same inputs. For example, we know that
Then
We do the same for the other values to produce [link].
The result is that the function
has been compressed vertically by
Each output value is divided in half, so the graph is half the original height.
2 | 4 | 6 | 8 | |
9 | 12 | 15 | 0 |
The graph in [link] is a transformation of the toolkit function
Relate this new function
to
and 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 that
With the basic cubic function at the same input,
Based on that, it appears that the outputs of
are
the outputs of the function
because
From this we can fairly safely conclude that
We can write a formula for
by 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.
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
the form
results in a horizontal stretch or compression. Consider the function
Observe [link]. The graph of
is a horizontal stretch of the graph of the function
by a factor of 2. The graph of
is a horizontal compression of the graph of the function
by a factor of 2.
Given a function
a new function
where
is a constant, is a horizontal stretch or horizontal compression of the function
then the graph will be compressed by
then the graph will be stretched by
then 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.
where
for a compression or
for a stretch.
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 [link] 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.
A function
is given as [link]. Create a table for the function
2 | 4 | 6 | 8 | |
1 | 3 | 7 | 11 |
The formula
tells us that the output values for
are the same as the output values for the function
at an input half the size. Notice that we do not have enough information to determine
because
and we do not have a value for
in our table. Our input values to
will need to be twice as large to get inputs for
that we can evaluate. For example, we can determine
We do the same for the other values to produce [link].
4 | 8 | 12 | 16 | |
1 | 3 | 7 | 11 |
[link] shows the graphs of both of these sets of points.
Because each input value has been doubled, the result is that the function
has been stretched horizontally by a factor of 2.
The graph of
looks like the graph of
horizontally compressed. Because
ends at
and
ends at
we can see that the
values have been compressed by
because
We might also notice that
and
Either way, we can describe this relationship as
This 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 of
in 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.
so using the square root function we get
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
which transformation should we start with? The answer here follows nicely from the order of operations. Given the output value of
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
for example, we have to think about how the inputs to the function
relate to the inputs to the function
Suppose we know
What input to
would produce that output? In other words, what value of
will allow
We would need
To solve for
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.
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 form
first vertically stretch by
and then vertically shift by
When combining horizontal transformations written in the form
first horizontally shift by
and then horizontally stretch by
When combining horizontal transformations written in the form
first horizontally stretch by
and then horizontally shift by
Horizontal and vertical transformations are independent. It does not matter whether horizontal or vertical transformations are performed first.
Given [link] for the function
create a table of values for the function
6 | 12 | 18 | 24 | |
10 | 14 | 15 | 17 |
There are three steps to this transformation, and we will work from the inside out. Starting with the horizontal transformations,
is a horizontal compression by
which means we multiply each
value by
See [link].
2 | 4 | 6 | 8 | |
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. See [link].
2 | 4 | 6 | 8 | |
20 | 28 | 30 | 34 |
Finally, we can apply the vertical shift, which will add 1 to all the output values. See [link].
2 | 4 | 6 | 8 | |
21 | 29 | 31 | 35 |
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 the
on 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 [link].
Next, we horizontally shift left by 2 units, as indicated by
See [link].
Last, we vertically shift down by 3 to complete our sketch, as indicated by the
on the outside of the function. See [link].
Access this online resource for additional instruction and practice with transformation of functions.
Vertical shift | (up for) |
Horizontal shift | (right for) |
Vertical reflection | |
Horizontal reflection | |
Vertical stretch | ( ) |
Vertical compression | |
Horizontal stretch | |
Horizontal compression | () |
axis. A graph can be reflected vertically by multiplying the output by –1.
axis. A graph can be reflected horizontally by multiplying the input by –1.
axis, whereas odd functions are symmetric about the origin.
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?
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?
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?
For a function
substitute
for
in
Simplify. If the resulting function is the same as the original function,
then the function is even. If the resulting function is the opposite of the original function,
then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.
Write a formula for the function obtained when the graph of
is shifted up 1 unit and to the left 2 units.
Write a formula for the function obtained when the graph of
is shifted down 3 units and to the right 1 unit.
Write a formula for the function obtained when the graph of
is shifted down 4 units and to the right 3 units.
Write a formula for the function obtained when the graph of
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
The graph of
is a horizontal shift to the left 43 units of the graph of
The graph of
is a horizontal shift to the right 4 units of the graph of
The graph of
is a vertical shift up 8 units of the graph of
The graph of
is a vertical shift down 7 units of the graph of
The graph of
is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of
For the following exercises, determine the interval(s) on which the function is increasing and decreasing.
decreasing on
and increasing on
decreasing on
For the following exercises, use the graph of
shown in [link] 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 functions
and
are given below. Write
and
as transformations of
−2 | −1 | 0 | 1 | 2 | |
−2 | −1 | −3 | 1 | 2 |
−1 | 0 | 1 | 2 | 3 | |
−2 | −1 | −3 | 1 | 2 |
−2 | −1 | 0 | 1 | 2 | |
−1 | 0 | −2 | 2 | 3 |
Tabular representations for the functions
and
are given below. Write
and
as transformations of
−2 | −1 | 0 | 1 | 2 | |
−1 | −3 | 4 | 2 | 1 |
−3 | −2 | −1 | 0 | 1 | |
−1 | −3 | 4 | 2 | 1 |
−2 | −1 | 0 | 1 | 2 | |
−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.
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.
even
odd
even
For the following exercises, describe how the graph of each function is a transformation of the graph of the original function
The graph of
is a vertical reflection (across the
-axis) of the graph of
The graph of
is a vertical stretch by a factor of 4 of the graph of
The graph of
is a horizontal compression by a factor of
of the graph of
The graph of
is a horizontal stretch by a factor of 3 of the graph of
The graph of
is a horizontal reflection across the
-axis and a vertical stretch by a factor of 3 of the graph of
For the following exercises, write a formula for the function
that results when the graph of a given toolkit function is transformed as described.
The graph of
is reflected over the
-axis and horizontally compressed by a factor of
.
The graph of
is reflected over the
-axis and horizontally stretched by a factor of 2.
The graph of
is vertically compressed by a factor of
then shifted to the left 2 units and down 3 units.
The graph of
is vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.
The graph of
is vertically compressed by a factor of
then shifted to the right 5 units and up 1 unit.
The graph of
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.
The graph of the function
is shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units.
The graph of
is stretched vertically by a factor of 2, shifted horizontally 4 units to the right, reflected across the horizontal axis, and then shifted vertically 3 units up.
The graph of the function
is compressed vertically by a factor of
The graph of the function is stretched horizontally by a factor of 3 and then shifted vertically downward by 3 units.
The graph of
is shifted right 4 units and then reflected across the vertical line
For the following exercises, use the graph in [link] to sketch the given transformations.
and is symmetric about the
axis
and is symmetric about the origin
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