Transformation of Functions in Precalculus
These are my notes on transformation of functions in precalculus.
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 \(g(x)=f(x)+k\), the function f(x) is shifted
vertically k units. To help visualize the concept of a vertical shift, consider
that \(y=f(x)\). Therefore, \(f(x)+k\) is equivalent to \(y+k\). Every unit of
y is replaced by \(y+k\), so the y-value increases or decreases depending on
the value of k. The result is a shift upward or downward.
Vertical Shift
Given a function, f(x), a new function g(x)=f(x)+k where k is a constant, is
a vertical shift of the function f(x). All the output values change by k units.
If k is positive, the graph will shift up. If k is negative, the graph will
shift down.
Example 1
Adding a Constant to a Function
To regulate temperature in a green building, airflow vents near the roof open
and close throughout the day. 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.
Solution:
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. Notice that for each input value, the output value has
increased by 20, so if we call the new function S(t), we could write
\(S(t)=V(t)+20\).
This notation tells us that, for any value of t, S(t) can be found by
evaluating the function V at the same input and then adding 20 to the result.
This defines S as a transformation of the function V, 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.
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.
Example 2
Shifting a Tabular Function Vertically
Solution:
The formula g(x)=f(x)-3 tells us that we can find the output values of g by
subtracting 3 from the output values of f.
\(f(2)=1\)
\(g(x)=f(x)-3\)
\(g(2)=f(2)-3\)
\(1-3=-2\)
As with the earlier vertical shift, notice the input values stay the same and
only the output values change.
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.
For example, if \(f(x)=x^2\). then \(g(x)=(x-2)^2\) 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 f.
Horizontal Shift
Given a function, f, a new function g(x)=f(x-h), where h is a constant, is a
horizontal shift of the function f. If h is positive the graph will shift to
the right. If h is negative then the graph will shift to the left.
Example 3
Adding a Constant to an Input
Returning to our building airflow example, suppose that in autumn the
facilities manager decides that the original venting plan starts too late, and
wants to begin the entire venting program 2 hours earlier. Sketch a graph
of the new function.
Solution:
We can set V(t) to be the original program and F(t) to be the revised program.
V(t)= the original venting plan
F(t)= starting 2 hours sooner
in the new graph, at each time, the airflow is the same as the original
function V, was 2 hours later. For example, in the original function V, the
airflow starts to change at 8 am, whereas for the function F, the airflow
starts to change at 6 am. The comparable function values are V(8)=F=(6).
Notice also that the vents first opened top \(220 ft^2\) at 10 am under the
original plan, while under the new plan the vents reach \(220 ft^2\) at
8am, so V(10)=F(8). In both cases, we see that,. because F(t) starts 2
hours sooner, h=-2. That means that the same output values are reached when:
\(F(t)=V(t)-(-2)) = V(t+2)\)
Note that V(t+2) has the effect of shifting the graph to the left. Horizontal
changes affect the domain of a function instead of the range and often seem
counterintuitive. The new function F(t) uses the same outputs as V(t), but
matches those outputs to inputs 2 hours earlier than those of V(t). Said
another way, we must add 2 hours to the input of V to find the corresponding
output of F: F(t)=V(t+2).
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.
Example 4
Shifting a Tabular Function Horizontally
A function f(x) is given, create a table for the function g(x)=f(x-3).
Solution:
The formula \(g(x)=f(x-3)\) tells us that the output values of g are the same
as the output value of f when the input value is 3 less than the original
value. for example, we know that f(2)=1. To get the same output from the
function g, we will need an input value that is 3 larger. We input a value
that is 3 larger for g(x) because the function takes 3 away before evaluating
the function f.
\(g(5)=f(5-3)=f(2)=1\)
The result is that the function g(x) has been shifted to the right by 3. Notice
the output values for g(x) remain the same as the output values for f(x),
but the corresponding values, x, 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.
Example 5
Identifying a Horizontal Shift of a Toolkit function
The graph represents a transformation of the toolkit function \(f(x)=x^2\).
Relate this new function g(x) to f(x), and then find a formula for g(x).
Solution:
Notice that the graph is identical in shape to the \(f(x)=x^2\) 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.
\(g(x)=f(x-2)\)
Notice how we must input the value x=2 to get the output value y=0. 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 f(x) function to write a formula for g(x)
by evaluating f(x-2).
\(F(x)=x^2\)
\(g(x)=f(x-2)\)
\(g(x)=f(x)-20=(x-2)^2\)
To determine whether the shift is +2 or -2, consider a single reference point on
the graph. For a quadratic, looking at the vertex point is convenient. In
the original function, f(0)=0. In out shifted function, g(2)=0. To
obtain the output value 0 from the function f, we need to decide whether a plus
or minus sign will work to satisfy g(2)=f(x-2)=f(0)=0. For this to work, we will
need to subtract 2 units form out input values.
Example 6
Interpreting Horizontal versus Vertical Shifts
The function G(m) gives the number of gallons of gas required to drive m
miles. Interpret G(m)+10 and G(m+10).
Solution:
G(m)+10 can be interpreted as adding 10 to the output, gallons. This is the gas
required to drive m miles, plus another 10 gallons of gas. The graph would
indicate a vertical shift.
G(m+10) 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 m miles. The
graph would indicate a horizontal shift.
Combining Vertical and Horizontal Shifts
Now that we have two transformations, we can combine them together. Vertical
shifts are outside changes that affect the input x 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.
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.
Example 7
Graphing Combined Vertical and Horizontal Shifts
Given f(x)=|x|, sketch a graph of h(x)=f(x+1)-3
Solution:
The function f is our toolkit absolute value function. We know that this graph
has a v shape, with the point at the origin. The graph of h has transformed f in
two ways: f(x+1) is a change on the inside of the function, giving a
horizontal shift left by 1, and the subtraction by 3 in f(x+1)-3 is a change
to the outside of the function, giving a vertical shift down by 3.
Example 8
Identifying Combined Vertical and Horizontal Shifts
Write a formula for the graph shown, which is a transformation of the
toolkit square root function.
Solution:
The graph of the toolkit function starts at the origin, so this graph has
been shifted 1 to the right and up 2. In function notation, we could write
that as \(h(x)=f(x-2)+2\). Using the formula for the square root function, we
can write \(h(x)=\sqrt{x-1}+2\).
Note that this transformation has changed the domain and range of the
function. This new graph has domain \([1,\infty)\) and range \([2,\infty)\).
Graphing Functions Using Reflections about the Axes
Notice that the vertical reflection produces a new graph that is a mirror
image of the base or original graph about the y-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 f(x), a new function g(x)=-f(x) is a vertical reflection of
the function f(x), sometimes called a reflection about the x-axis.
Given a function f(x), a new function g(x)=f(-x) is a horizontal reflection of
the function f(x), sometimes called a reflection about the y-axis.
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 outputs by -1 for a horizontal reflection. The new graph
is a reflection of the original graph about the y-axis.
Example 9
Reflecting a Graph Horizontally and Vertically
Reflect the graph of \(s(t)=\sqrt{t}\) vertically and horizontally.
Solution:
Reflecting the graph vertically means that each output value will be reflected
over the horizontal t-axis.
Because each output value is the opposite of the original output value, we
can write: \(V(t)=-s(t)\).
Notice that his is an outside change, or vertical shift, that affects the
output s(t) values, so the negative sign belongs outside of the function.
Reflecting horizontally means that each input value will be reflected over
the vertical axis.
Because each input value is the opposite of the original input value, we
can write: \(H(t)=s(-t)\).
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 transformation can affect the domain and range of the
functions. While the original square root function has domain \([0,\infty)\) and
range \([0,\infty), the vertical reflection gives the V(t) function the range
\((-\infty.0])\) and the horizontal reflection gives the H(t) function the
domain \((-\infty,0])\).
Example 10
Reflecting a Tabular Function Horizontally and Vertically
A function f(x) is given, create a table for the functions:
g(x)=-f(x)
h(x)=f(-x)
x 2 4 6 8
f(x)1 3 7 11
Solution:
For g(x), 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.
For h(x), the negative sign inside the function indicates a horizontal
reflection, so each input value will be the original input value and the h(x)
values stay the same as the f(x) values.
Example 11
Applying a Learning Model Equation
A common model for learning has an equation similar to \(k(t)=2^{-2} + 1\),
where k is the percentage of mastery that can be achieved after t
practice sessions. This is a transformation of the function \(f(t)=2^t\).
Solution:
This equation combines three transformations into one equation.
1. A horizontal reflection:\(f(-t)=2^{-t}\)
2. A vertical reflection:\(-f(-t)=-2^{-t}\)
3. A vertical shift:\(-f(-t)=-2^{-t} + 1\)
We can sketch a graph by applying these transformations one at a
time to the original function. Let us follow two points through each of
the three transformations. We will choose the points (0,1) and (1,2).
1. First, we apply a horizontal reflection: (0,1)(-1,2)
2. Then, we apply a vertical reflection:(0,-1)(-1,-2)
3. 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.
As a model for learning, this function would be limited to a domain of
\(t \geq 0\), with a corresponding range of [0,1).
Determining Even and Odd Functions
Some functions exhibit symmetry so that reflections result in the original
graph. For example, horizontally reflecting the toolkit functions
\(f(x)=x^2\) or \(f(x)=|x|\) will result in the original graph. We say
these types of graphs are symmetric about the y-axis. Functions whose graphs
are symmetric about the y-axis are called even functions. In the graphs
of \(f(x)=x^3\) or \(f(x)=1/x\) were reflected over both axes, the result
would be the original graph.
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. A function
can be neither even nor odd if it does not exhibit symmetry. For example,
\(f(x)=2^x\) is neither even nor odd. Also, the only function that is both
even and odd is the constant function \(f(x)=0\).
Even and Odd Functions
A function is called an even function if for every input of x:
\(f(x)=f(-x)\).
The graph of an even function is symmetric about the y-axis.
A function is called an odd function if for every input of x:
\(f(x)=-f(-x)\).
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.
1. Determine whether the function satisfies \(f(x)=f(-x)\). If it does, it is
even.
2. Determine whether the function satisfies \(f(x)=-f(-x)\). If it does, it
is odd.
3. If the function does not satisfy either rule, it is neither even nor odd.
Example 12
Determining whether a Function is Even, odd, or neither
Is the function \(f(x)=x^3+2x\) even, odd, or neither?
Solution:
Without looking at a graph, we can determine whether the function is even or
odd by finding formulas for the reflections and determining if they
return us to the original function. Let us begin with the rule for even
functions.
\(f(-x)=(-x)^3 + 2(-x) = -x^3 - 2x\)
This does not return us to the original function, so this function is not
even. We can now test the rule for odd functions.
\(-f(-x)=-(-x^3-2x)=x^3+2x\)
Because \(-f(-x)=f(x)\), this is an odd function.
Consider the graph of f. Notice that the graph is symmetric about the
origin. For every point (x,y) on the graph, the corresponding
point(-x,-y) is also on the graph. For example, (1,3) is on the graph of f, and
the corresponding point (-1,-3) is also on the graph.
Graphing Functions Using Stretches and Compressions
Adding a constant to the inputs 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.
Vertical Stretches and Compressions
Given a function f(x), a new function g(x)=af(x), where a is a constant, is
a vertical stretch or vertical compression of the function f(x).
1. If a>1 then the graph will be stretched.
2. If 0<a<1, then the graph will be compressed.
3. If a<0, then there will be a combination of a vertical stretch with a
vertical reflection.
Given a function, graph its vertical stretch
1. Identify the value of a
2. Multiply all range values by a
3. If a>1, the graph is stretched by a factor of a
If 0<a(1, the graph is compressed by a factor of a
If a<0, the graph is either stretched or compressed and also reflected about
the x-axis.
Example 13
Graphing a Vertical Stretch
A function P(t) models the population of fruit flies.
A scientist is comparing this population to another population, Q, whose
growth follows the same pattern, but is twice as large. Sketch a graph of
this population.
Solution:
Because the population is always twice as large, the new population's output
values are always twice the original function's output values.
If we choose four reference points, (0,1)(3,3)(6,2)(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.
This means that for any input t, the value of the function Q is twice the
value of the function P. 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, t, 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.
1. Determine the value of a
2. Multiply all of the output values by a
Example 14
Finding a Vertical Compression of a Tabular Function
A function f is given, create a table for the function \(g(x)=\frac{1}{2}f(x)\).
x 2 4 6 8
f(x) 1 3 7 11
Solution:
The formula \(g(x)=\frac{1}{2}f(x)\) tells us that the output values of g are
half of the output values of f with the same inputs. For example, we know that
f(4)=3. Then
\(g(4)=\frac{1}{2}f(4)=\frac{1}{2}(3)=\frac{3}{2}\)
The result is that the function g(x) has been compressed vertically by
1/2. Each output value is divided in half, so the graph is half the
original height.
Example 15
Recognizing a Vertical Stretch
The graph shown is a transformation of the toolkit function \(f(x)=x^3\).
Relate this new function g(x) to f(x), and then find a formula for g(x).
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
g(2)=2. With the basic cubic function at the same input, \(f(2)=2^3=8\).
based on that, it appears that the outputs of g are 1/4 the outputs of the
function f because g(2)=1/4 f92). From this we can safely conclude that
g9x)=1/4f(x). We can write a formula for g by using the definition of f.
\(g(x)=\frac{1}{4}f(x)=\frac{1}{4}x^3\)
Horizontal Stretches and Compressions
Now we consider changes to the inside of the 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 y=f(x), the form y=f(bx) results in a horizontal stretch or
compression. Consider the function \(y=x^2\). The graph of \(y=(0.5x)^2\) is a
horizontal stretch of the graph of the function \(y=x^2\) by a factor of 1/2.
Horizontal Stretches and Compressions
Given a function f(x), a new function g(x)=f(bx), where b is a constant, and is
a horizontal stretch or horizontal compression of the function f(x).
1. If b>1, then the graph will be compressed 1/b.
2. If 0<b<1, then the graph will be stretched by 1/b.
3. If b<0<1, then there will be combination of a horizontal stretch or
horizontal compression with a horizontal reflection.
Given a description of a function, sketch a horizontal compression or stretch.
1. Write a formula to represent the function
2. Set g(x)=f(bx) where b>1 for a compression or 0<b<1 for a stretch.
Example 16
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:
\(R(1)=P(2)\)
\(R(2)=P(4)\)
\(R(t)=P(2t)\)
Note that the effect on the graph is a horizontal compression where all input
values are half of their original distance from the vertical axis.
Example 17
Finding a Horizontal Stretch for a Tabular Function
A function f(x) is given, create a table for the function g(x)=f(1/2x).
x 2 4 6 8
f(x) 1 3 7 11
Solution:
The formula g(x)=f(1/2x) tells us that the output values for g are the same as
the output values for the function f at an input half the size. Notice that we
do not have enough information to determine g(2) because g(2)-f(1/2(2))=f(1),
and we do not have a value for f(1) in our table. Our input values to g will
need to be twice as large to get inputs for f that we can evaluate. For example,
we can determine g(4).
\(g(4)=f(1/2(4))=f(2)=1\)
Because each input value has been doubled, the result is that the function g(x)
has been stretched horizontally by a factor of 2.
Example 18
Recognizing a Horizontal Compression on a Graph
Relate the function g(x) to f(x).
Solution:
The graph of g(x) looks like the graph of f(x) horizontally compressed. Because
f(x) ends at (6,4) and g(x) ends at (2,4), we can see that the x-values have
been compressed to 1/3, because 6(1/3)=2. We might also notice that g(2)=f(6)
and g(1)=f(3). Either way, we can describe this relationship as g(x)=f(3x). This
is a horizontal compression by 1/3.
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 1/4 in our
function: f(1/4(x)). This means that the input values must be 4 times larger to
produce the same result, requiring the input to be larger, causing the
horizontal stretching.
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 2f(x)+3, which transformation should we start
with? The answer here follows nicely from the order of operations. Given the
output value of f(x), 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
g(x)=f(2x+3), for example, we have to think about how the inputs to the function
g relate to the inputs to the function f. Suppose we know f(7)=12. What input to
g would produce that output? In other words, what value of x will allow
g(x)=f(2x+3)=12? We would need 2x+3=7. To solve for x, 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.
\(f(bx+p)=f(b(x+\frac{p}{b}))\)
So:
\(f(x)=(2x+4)^2\)
We can factor out a 2
\(f(x)=(2(x+2))^2\)
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.
Combining Transformations
When combining vertical transformations written in the form af(x)+k, first
vertically stretch by a and then vertically shift by k.
When combining horizontal transformations written in the form f(bx-h), first
horizontally shift by h/b and then horizontally stretch by 1/b.
When combining horizontal transformations written in the form f(b(x-h)), first
horizontally stretch by 1/b and then horizontally shift by h.
Horizontal and vertical transformation are independent. It does not matter
whether horizontal or vertical transformations are performed first.
Example 19
Finding a Triple Transformation of a Tabular Function
For the function f(x), create a table of values for the function g(x)=2f(3x)+1.
x 6 12 18 24
f(x) 10 14 15 17
Solution:
There are three steps to this transformation, and we will work from the inside
out. Starting with the horizontal transformations, f(3x) is a horizontal
compression by 1/3, which means we multiply each x-value by 1/3.
x 2 4 6 8
f(x) 10 14 15 17
Looking now to the vertical transformations, we start with the vertical
stretch, which will multiply the output values by 2. We apply this to the
previous transformation.
x 2 4 6 8
2f(3x) 20 28 30 34
Finally, we can apply the vertical shift, which will add 1 to all the output
values.
x 2 4 6 8
g(x)=2f(3x)+1 21 29 31 35
Example 20
Finding a Triple Transformation of a Graph
Use the graph of f(x) to sketch a graph of k(x)=f(1/2x+1)-3
Solution:
To simplify, let us start by factoring out the inside of the function.
\(f(\frac({1}{2}x+1)-3=f(\frac{1}{2}(x+2)-3\)
By factoring the inside, we can first horizontally stretch by 2, as indicated by
the 1/2 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).
Next, we horizontally shift left by 2 units, as indicated by x+2.
Last, we vertically shift down by 3 to complete our sketch, as indicated by the
-3 on the outside of the function.