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 realworld applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a realworld 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[latex]\,b\,[/latex]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) = b^{x}
An exponential function with the form[latex]\,b>0,[/latex]has these characteristics:
 onetoone function
 horizontal asymptote:
 domain:
 range:
 xintercept: none
 yintercept:
 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.
 Create a table of points.
 Plot at leastpoint from the table, including the yintercept
 Draw a smooth curve through the points.
 State the domain,the range,and the horizontal asymptote,
Sketching the Graph of an Exponential Function of the Form f(x) = b^{x}
Sketch a graph ofState the domain, range, and asymptote.
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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 yintercept,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
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Try It
Sketch the graph ofState the domain, range, and asymptote.
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The domain isthe range isthe horizontal asymptote is
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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 yintercept shifts upunits to
 The asymptote shifts upunits to
 The range becomes
 When the function is shifted downunits to
 The yintercept 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 yintercept shifts such that:
 When the function is shifted leftunits tothe yintercept becomesThis is becauseso the initial value of the function is
 When the function is shifted rightunits tothe yintercept becomesAgain, see thatso the initial value of the function is
Shifts of the Parent Function f(x) = b^{x}
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 yintercept becomes
 The horizontal asymptote becomes
 The range becomes
 The domain,remains unchanged.
How To
Given an exponential function with the formgraph the translation.
 Draw the horizontal asymptote
 Identify the shift asShift the graph ofleftunits ifis positive, and rightunits ifis negative.
 Shift the graph ofupunits ifis positive, and downunits ifis negative.
 State the domain,the range,and the horizontal asymptote
Graphing a Shift of an Exponential Function
GraphState the domain, range, and asymptote.
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We have an exponential equation of the form with[latex]\,c=1,[/latex] 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.
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Try It
GraphState domain, range, and asymptote.
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The domain isthe range isthe horizontal asymptote is
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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 “Y_{1}=”.
 Enter the given value forin the line headed “Y_{2}=”.
 Press [WINDOW]. Adjust the yaxis so that it includes the value entered for “Y_{2}=”.
 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.
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Press [Y=] and enternext to Y_{1}=. 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 xcoordinate 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,
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Try It
Solvegraphically. Round to the nearest thousandth.
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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).
Graphing Reflections
In addition to shifting, compressing, and stretching a graph, we can also reflect it about the xaxis or the yaxis. When we multiply the parent functionbywe get a reflection about the xaxis. When we multiply the input bywe get a reflection about the yaxis. For example, if we begin by graphing the parent function we can then graph the two reflections alongside it. The reflection about the xaxis,is shown on the left side of (Figure), and the reflection about the yaxis is shown on the right side of (Figure).
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 xintercept 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 xintercept 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


Stretch and Compress


Reflect about the xaxis  
Reflect about the yaxis  
General equation for all translations 
Translations of Exponential Functions
A translation of an exponential function has the form
Where the parent function,[latex]\,b>1,[/latex]is
 shifted horizontallyunits to the left.
 stretched vertically by a factor ofif
 compressed vertically by a factor ofif
 shifted verticallyunits.
 reflected about the xaxis 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 yaxis, and then shifted upunits.
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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 yaxis. We replacewithto get:
 The graph is shifted vertically 4 units, so
Substituting in the general form we get,
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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 xaxis and then shifted down units.
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the domain isthe range isthe horizontal asymptote is
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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 yintercept 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 xaxis. When the input is multiplied bythe result, is a reflection about the yaxis. 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?
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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.
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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 yaxis and stretched vertically by a factor ofWhat is the equation of the new function,State its yintercept, domain, and range.
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yintercept:Domain: all real numbers; Range: all real numbers greater than
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The graph ofis reflected about the yaxis and compressed vertically by a factor ofWhat is the equation of the new function,State its yintercept, domain, and range.
The graph ofis reflected about the xaxis and shifted upwardunits. What is the equation of the new function,State its yintercept, domain, and range.
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yintercept:Domain: all real numbers; Range: all real numbers less than
[/hiddenanswer]
The graph ofis shifted rightunits, stretched vertically by a factor ofreflected about the xaxis, and then shifted downwardunits. What is the equation of the new function,State its yintercept (to the nearest thousandth), domain, and range.
The graph of is shifted leftunits, stretched vertically by a factor ofreflected about the xaxis, and then shifted downwardunits. What is the equation of the new function,State its yintercept, domain, and range.
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yintercept:Domain: all real numbers; Range: all real numbers greater than
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Graphical
For the following exercises, graph the function and its reflection about the yaxis on the same axes, and give the yintercept.
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yintercept:
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For the following exercises, graph each set of functions on the same axes.
[latex]g\left(x\right)=3{\left(2\right)}^{x},[/latex]and
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[latex]g\left(x\right)=2{\left(3\right)}^{x},[/latex]and
For the following exercises, match each function with one of the graphs in (Figure).
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B
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A
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E
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For the following exercises, use the graphs shown in (Figure). All have the form
Which graph has the largest value for
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D
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Which graph has the smallest value for
Which graph has the largest value for
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C
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Which graph has the smallest value for
For the following exercises, graph the function and its reflection about the xaxis on the same axes.
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For the following exercises, graph the transformation ofGive the horizontal asymptote, the domain, and the range.
[revealanswer q=”979418″]Show Solution[/revealanswer]
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Horizontal asymptote: Domain: all real numbers; Range: all real numbers strictly greater than
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For the following exercises, describe the end behavior of the graphs of the functions.
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;[/hiddenanswer]
[revealanswer q=”fsid1165135543075″]Show Solution[/revealanswer]
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As ,
;[/hiddenanswer]
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
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Shift2 units left
Shift5 units right
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Reflectabout the xaxis
Reflectabout the yaxis
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For the following exercises, each graph is a transformation ofWrite an equation describing the transformation.
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For the following exercises, find an exponential equation for the graph.
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Numeric
For the following exercises, evaluate the exponential functions for the indicated value of
for
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for
for
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Technology
For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth.
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Extensions
Explore and discuss the graphs ofandThen make a conjecture about the relationship between the graphs of the functionsandfor any real number
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The graph ofis the refelction about the yaxis of the graph ofFor any real numberand functionthe graph ofis the the reflection about the yaxis,
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Prove the conjecture made in the previous exercise.
Explore and discuss the graphs of[latex]\,g\left(x\right)={4}^{x2},[/latex]andThen make a conjecture about the relationship between the graphs of the functionsandfor any real number n and real number
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The graphs ofandare the same and are a horizontal shift to the right of the graph ofFor any real number n, real number and function the graph ofis the horizontal shift
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Prove the conjecture made in the previous exercise.