# 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.