Exponential and Logarithmic Functions

# Graphs of Exponential Functions

### Learning Objectives

• Graph exponential functions.
• Graph exponential functions using transformations.

As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool. It gives us another layer of insight for predicting future events.

### Graphing Exponential Functions

Before we begin graphing, it is helpful to review the behavior of exponential growth. Recall the table of values for a function of the formwhose base is greater than one. We’ll use the functionObserve how the output values in (Figure) change as the input increases by

Each output value is the product of the previous output and the base,We call the basethe constant ratio. In fact, for any exponential function with the form$\,b\,$is the constant ratio of the function. This means that as the input increases by 1, the output value will be the product of the base and the previous output, regardless of the value of

Notice from the table that

• the output values are positive for all values of
• asincreases, the output values increase without bound; and
• asdecreases, the output values grow smaller, approaching zero.

(Figure) shows the exponential growth function

The domain ofis all real numbers, the range is and the horizontal asymptote is

To get a sense of the behavior of exponential decay, we can create a table of values for a function of the formwhose base is between zero and one. We’ll use the functionObserve how the output values in (Figure) change as the input increases by

Again, because the input is increasing by 1, each output value is the product of the previous output and the base, or constant ratio

Notice from the table that

• the output values are positive for all values of
• asincreases, the output values grow smaller, approaching zero; and
• asdecreases, the output values grow without bound.

(Figure) shows the exponential decay function,

The domain ofis all real numbers, the range isand the horizontal asymptote is

### Characteristics of the Graph of the Parent Function f(x) = bx

An exponential function with the form$\,b>0,$has these characteristics:

• one-to-one function
• horizontal asymptote:
• domain:
• range:
• x-intercept: none
• y-intercept:
• increasing if
• decreasing if

(Figure) compares the graphs of exponential growth and decay functions.

### How To

Given an exponential function of the formgraph the function.

1. Create a table of points.
2. Plot at leastpoint from the table, including the y-intercept
3. Draw a smooth curve through the points.
4. State the domain,the range,and the horizontal asymptote,

### Sketching the Graph of an Exponential Function of the Form f(x) = bx

Sketch a graph ofState the domain, range, and asymptote.

Before graphing, identify the behavior and create a table of points for the graph.

• Sinceis between zero and one, we know the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote
• Create a table of points as in (Figure).
• Plot the y-intercept,along with two other points. We can useand

Draw a smooth curve connecting the points as in (Figure).

The domain isthe range isthe horizontal asymptote is

### Try It

Sketch the graph ofState the domain, range, and asymptote.

The domain isthe range isthe horizontal asymptote is

### Graphing Transformations of Exponential Functions

Transformations of exponential graphs behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations—shifts, reflections, stretches, and compressions—to the parent functionwithout loss of shape. For instance, just as the quadratic function maintains its parabolic shape when shifted, reflected, stretched, or compressed, the exponential function also maintains its general shape regardless of the transformations applied.

#### Graphing a Vertical Shift

The first transformation occurs when we add a constantto the parent function giving us a vertical shiftunits in the same direction as the sign. For example, if we begin by graphing a parent function, we can then graph two vertical shifts alongside it, usingthe upward shift,and the downward shift,Both vertical shifts are shown in (Figure).

Observe the results of shiftingvertically:

• The domain,remains unchanged.
• When the function is shifted upunits to
• The y-intercept shifts upunits to
• The asymptote shifts upunits to
• The range becomes
• When the function is shifted downunits to
• The y-intercept shifts downunits to
• The asymptote also shifts downunits to
• The range becomes

#### Graphing a Horizontal Shift

The next transformation occurs when we add a constantto the input of the parent function giving us a horizontal shiftunits in the opposite direction of the sign. For example, if we begin by graphing the parent function we can then graph two horizontal shifts alongside it, usingthe shift left, and the shift right,Both horizontal shifts are shown in (Figure).

Observe the results of shiftinghorizontally:

• The domain,remains unchanged.
• The asymptote,remains unchanged.
• The y-intercept shifts such that:
• When the function is shifted leftunits tothe y-intercept becomesThis is becauseso the initial value of the function is
• When the function is shifted rightunits tothe y-intercept becomesAgain, see thatso the initial value of the function is

### Shifts of the Parent Function f(x) = bx

For any constantsandthe functionshifts the parent function

• verticallyunits, in the same direction of the sign of
• horizontallyunits, in the opposite direction of the sign of
• The y-intercept becomes
• The horizontal asymptote becomes
• The range becomes
• The domain,remains unchanged.

### How To

Given an exponential function with the formgraph the translation.

1. Draw the horizontal asymptote
2. Identify the shift asShift the graph ofleftunits ifis positive, and rightunits ifis negative.
3. Shift the graph ofupunits ifis positive, and downunits ifis negative.
4. State the domain,the range,and the horizontal asymptote

### Graphing a Shift of an Exponential Function

GraphState the domain, range, and asymptote.

We have an exponential equation of the form with$\,c=1,$ and

Draw the horizontal asymptote, so draw

Identify the shift as so the shift is

Shift the graph ofleft 1 units and down 3 units.

The domain isthe range isthe horizontal asymptote is[/hidden-answer]

### Try It

GraphState domain, range, and asymptote.

The domain isthe range isthe horizontal asymptote is

### How To

Given an equation of the formfor use a graphing calculator to approximate the solution.

• Press [Y=]. Enter the given exponential equation in the line headed “Y1=”.
• Enter the given value forin the line headed “Y2=”.
• Press [WINDOW]. Adjust the y-axis so that it includes the value entered for “Y2=”.
• Press [GRAPH] to observe the graph of the exponential function along with the line for the specified value of
• To find the value ofwe compute the point of intersection. Press [2ND] then [CALC]. Select “intersect” and press [ENTER] three times. The point of intersection gives the value of x for the indicated value of the function.

### Approximating the Solution of an Exponential Equation

Solvegraphically. Round to the nearest thousandth.

Press [Y=] and enternext to Y1=. Then enter 42 next to Y2=. For a window, use the values –3 to 3 forand –5 to 55 forPress [GRAPH]. The graphs should intersect somewhere near

For a better approximation, press [2ND] then [CALC]. Select [5: intersect] and press [ENTER] three times. The x-coordinate of the point of intersection is displayed as 2.1661943. (Your answer may be different if you use a different window or use a different value for Guess?) To the nearest thousandth,

### Try It

Solvegraphically. Round to the nearest thousandth.

#### Graphing a Stretch or Compression

While horizontal and vertical shifts involve adding constants to the input or to the function itself, a stretch or compression occurs when we multiply the parent functionby a constantFor example, if we begin by graphing the parent functionwe can then graph the stretch, usingto getas shown on the left in (Figure), and the compression, usingto getas shown on the right in (Figure).

### Stretches and Compressions of the Parent Function f(x) = bx

For any factorthe function

• is stretched vertically by a factor ofif
• is compressed vertically by a factor ofif
• has a y-intercept of
• has a horizontal asymptote at a range of and a domain ofwhich are unchanged from the parent function.

### Graphing the Stretch of an Exponential Function

Sketch a graph ofState the domain, range, and asymptote.

Before graphing, identify the behavior and key points on the graph.

• Sinceis between zero and one, the left tail of the graph will increase without bound asdecreases, and the right tail will approach the x-axis asincreases.
• Sincethe graph ofwill be stretched by a factor of
• Create a table of points as shown in (Figure).
• Plot the y-intercept,along with two other points. We can useand

Draw a smooth curve connecting the points, as shown in (Figure).

The domain isthe range isthe horizontal asymptote is

### Try It

Sketch the graph ofState the domain, range, and asymptote.

The domain isthe range isthe horizontal asymptote is[/hidden-answer]

#### Graphing Reflections

In addition to shifting, compressing, and stretching a graph, we can also reflect it about the x-axis or the y-axis. When we multiply the parent functionbywe get a reflection about the x-axis. When we multiply the input bywe get a reflection about the y-axis. For example, if we begin by graphing the parent function we can then graph the two reflections alongside it. The reflection about the x-axis,is shown on the left side of (Figure), and the reflection about the y-axis is shown on the right side of (Figure).

### Reflections of the Parent Function f(x) = bx

The function

• reflects the parent functionabout the x-axis.
• has a y-intercept of
• has a range of
• has a horizontal asymptote atand domain ofwhich are unchanged from the parent function.

The function

• reflects the parent functionabout the y-axis.
• has a y-intercept of a horizontal asymptote at a range of and a domain of which are unchanged from the parent function.

### Writing and Graphing the Reflection of an Exponential Function

Find and graph the equation for a function,that reflectsabout the x-axis. State its domain, range, and asymptote.

Since we want to reflect the parent functionabout the x-axis, we multiplybyto get,Next we create a table of points as in (Figure).

Plot the y-intercept,along with two other points. We can useand

Draw a smooth curve connecting the points:

The domain isthe range isthe horizontal asymptote is[/hidden-answer]

### Try It

Find and graph the equation for a function, that reflectsabout the y-axis. State its domain, range, and asymptote.

The domain isthe range isthe horizontal asymptote is

#### Summarizing Translations of the Exponential Function

Now that we have worked with each type of translation for the exponential function, we can summarize them in (Figure) to arrive at the general equation for translating exponential functions.

1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept remains (1, 0), the key point changes to (b^(-1), 1), the domain remains (0, infinity), and the range remains (-infinity, infinity). The second column shows the left shift of the equation g(x)=log_b(x) when b>1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept changes to (-1, 0), the key point changes to (-b, 1), the domain changes to (-infinity, 0), and the range remains (-infinity, infinity).”>

Translations of the Parent Function
Translation Form
Shift

• Horizontallyunits to the left
• Verticallyunits up
Stretch and Compress

• Stretch if
• Compression if
General equation for all translations

### Translations of Exponential Functions

A translation of an exponential function has the form

Where the parent function,$\,b>1,$is

• shifted horizontallyunits to the left.
• stretched vertically by a factor ofif
• compressed vertically by a factor ofif
• shifted verticallyunits.
• reflected about the x-axis when

Note the order of the shifts, transformations, and reflections follow the order of operations.

### Writing a Function from a Description

Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.

• is vertically stretched by a factor of, reflected across the y-axis, and then shifted upunits.

We want to find an equation of the general formWe use the description provided to find and

• We are given the parent function so
• The function is stretched by a factor of, so
• The function is reflected about the y-axis. We replacewithto get:
• The graph is shifted vertically 4 units, so

Substituting in the general form we get,

The domain isthe range isthe horizontal asymptote is[/hidden-answer]

### Try It

Write the equation for function described below. Give the horizontal asymptote, the domain, and the range.

• is compressed vertically by a factor of reflected across the x-axis and then shifted down units.

the domain isthe range isthe horizontal asymptote is

Access this online resource for additional instruction and practice with graphing exponential functions.

### Key Equations

 General Form for the Translation of the Parent Function

### Key Concepts

• The graph of the functionhas a y-intercept atdomainrange and horizontal asymptoteSee (Figure).
• Ifthe function is increasing. The left tail of the graph will approach the asymptote and the right tail will increase without bound.
• If the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote
• The equationrepresents a vertical shift of the parent function
• The equationrepresents a horizontal shift of the parent functionSee (Figure).
• Approximate solutions of the equationcan be found using a graphing calculator. See (Figure).
• The equation where represents a vertical stretch ifor compression ifof the parent functionSee (Figure).
• When the parent functionis multiplied bythe result, is a reflection about the x-axis. When the input is multiplied bythe result, is a reflection about the y-axis. See (Figure).
• All translations of the exponential function can be summarized by the general equationSee (Figure).
• Using the general equation we can write the equation of a function given its description. See (Figure).

### Section Exercises

#### Verbal

What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph?

An asymptote is a line that the graph of a function approaches, aseither increases or decreases without bound. The horizontal asymptote of an exponential function tells us the limit of the function’s values as the independent variable gets either extremely large or extremely small.

What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraically?

#### Algebraic

The graph ofis reflected about the y-axis and stretched vertically by a factor ofWhat is the equation of the new function,State its y-intercept, domain, and range.

y-intercept:Domain: all real numbers; Range: all real numbers greater than

The graph ofis reflected about the y-axis and compressed vertically by a factor ofWhat is the equation of the new function,State its y-intercept, domain, and range.

The graph ofis reflected about the x-axis and shifted upwardunits. What is the equation of the new function,State its y-intercept, domain, and range.

y-intercept:Domain: all real numbers; Range: all real numbers less than

The graph ofis shifted rightunits, stretched vertically by a factor ofreflected about the x-axis, and then shifted downwardunits. What is the equation of the new function,State its y-intercept (to the nearest thousandth), domain, and range.

The graph of is shifted leftunits, stretched vertically by a factor ofreflected about the x-axis, and then shifted downwardunits. What is the equation of the new function,State its y-intercept, domain, and range.

y-intercept:Domain: all real numbers; Range: all real numbers greater than

#### Graphical

For the following exercises, graph the function and its reflection about the y-axis on the same axes, and give the y-intercept.

y-intercept:

For the following exercises, graph each set of functions on the same axes.

$g\left(x\right)=3{\left(2\right)}^{x},$and

$g\left(x\right)=2{\left(3\right)}^{x},$and

For the following exercises, match each function with one of the graphs in (Figure).

B

A

E

For the following exercises, use the graphs shown in (Figure). All have the form

Which graph has the largest value for

D

Which graph has the smallest value for

Which graph has the largest value for

C

Which graph has the smallest value for

For the following exercises, graph the function and its reflection about the x-axis on the same axes.

For the following exercises, graph the transformation ofGive the horizontal asymptote, the domain, and the range.

Horizontal asymptote: Domain: all real numbers; Range: all real numbers strictly greater than

For the following exercises, describe the end behavior of the graphs of the functions.

As ,

For the following exercises, start with the graph ofThen write a function that results from the given transformation.

Shift 4 units upward

Shift3 units downward

Shift2 units left

Shift5 units right

For the following exercises, each graph is a transformation ofWrite an equation describing the transformation.

For the following exercises, find an exponential equation for the graph.

#### Numeric

For the following exercises, evaluate the exponential functions for the indicated value of

for

for

for

#### Technology

For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth.

#### Extensions

Explore and discuss the graphs ofandThen make a conjecture about the relationship between the graphs of the functionsandfor any real number

The graph ofis the refelction about the y-axis of the graph ofFor any real numberand functionthe graph ofis the the reflection about the y-axis,

Prove the conjecture made in the previous exercise.

Explore and discuss the graphs of$\,g\left(x\right)={4}^{x-2},$andThen make a conjecture about the relationship between the graphs of the functionsandfor any real number n and real number