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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 form whose base is greater than one. We’ll use the function Observe how the output values in Table 1 change as the input increases by
Table 1
Each output value is the product of the previous output and the base, We call the base the constant ratio. In fact, for any exponential function with the form 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
- as increases, the output values increase without bound; and
- as decreases, the output values grow smaller, approaching zero.
Figure 1 shows the exponential growth function
Figure 1 Notice that the graph gets close to the x-axis, but never touches it.
The domain of is 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 form whose base is between zero and one. We’ll use the function Observe how the output values in Table 2 change as the input increases by
Table 2
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
- as increases, the output values grow smaller, approaching zero; and
- as decreases, the output values grow without bound.
Figure 2 shows the exponential decay function,
Figure 2
The domain of is all real numbers, the range is and the horizontal asymptote is
Characteristics of the Graph of the Parent Function
An exponential function with the form has these characteristics:
- one-to-one function
- horizontal asymptote:
- domain:
- range:
- x-intercept: none
- y-intercept:
- increasing if
- decreasing if
Figure 3 compares the graphs of exponential growth and decay functions.
Figure 3
How To
Given an exponential function of the form graph the function.
- Create a table of points.
- Plot at least point from the table, including the y-intercept
- Draw a smooth curve through the points.
- State the domain, the range, and the horizontal asymptote,
Example 1
Sketching the Graph of an Exponential Function of the Form f(x) = bx
Sketch a graph of State the domain, range, and asymptote.
- Answer
Before graphing, identify the behavior and create a table of points for the graph.
- Since is 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 Table 3.
Table 3
Draw a smooth curve connecting the points as in Figure 4.
Figure 4
The domain is the range is the horizontal asymptote is
Try It #1
Sketch the graph of State the domain, range, and asymptote.
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 function without 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 constant to the parent function giving us a vertical shift units 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, using the upward shift, and the downward shift, Both vertical shifts are shown in Figure 5.
Figure 5
Observe the results of shifting vertically:
- The domain, remains unchanged.
- When the function is shifted up units to
- The y-intercept shifts up units to
- The asymptote shifts up units to
- The range becomes
- When the function is shifted down units to
- The y-intercept shifts down units to
- The asymptote also shifts down units to
- The range becomes
Graphing a Horizontal Shift
The next transformation occurs when we add a constant to the input of the parent function giving us a horizontal shift units 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, using the shift left, and the shift right, Both horizontal shifts are shown in Figure 6.
Figure 6
Observe the results of shifting horizontally:
- The domain, remains unchanged.
- The asymptote, remains unchanged.
- The y-intercept shifts such that:
- When the function is shifted left units to the y-intercept becomes This is because so the initial value of the function is
- When the function is shifted right units to the y-intercept becomes Again, see that so the initial value of the function is
Shifts of the Parent Function f(x) = b x
For any constants and the function shifts the parent function
- vertically units, in the same direction of the sign of
- horizontally units, 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 form graph the translation.
- Draw the horizontal asymptote
- Identify the shift as Shift the graph of left units if is positive, and right units if is negative.
- Shift the graph of up units if is positive, and down units if is negative.
- State the domain, the range, and the horizontal asymptote
Example 2
Graphing a Shift of an Exponential Function
Graph State the domain, range, and asymptote.
- Answer
We have an exponential equation of the form with and
Draw the horizontal asymptote , so draw
Identify the shift as so the shift is
Shift the graph of left 1 units and down 3 units.
Figure 7
The domain is the range is the horizontal asymptote is
Try It #2
Graph State domain, range, and asymptote.
How To
Given an equation of the form for use a graphing calculator to approximate the solution.
- Press [Y=]. Enter the given exponential equation in the line headed “Y1=”.
- Enter the given value for in 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 of we 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.
Example 3
Approximating the Solution of an Exponential Equation
Solve graphically. Round to the nearest thousandth.
- Answer
Press [Y=] and enter next to Y1=. Then enter 42 next to Y2=. For a window, use the values –3 to 3 for and –5 to 55 for Press [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 #3
Solve graphically. 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 function by a constant For example, if we begin by graphing the parent function we can then graph the stretch, using to get as shown on the left in Figure 8, and the compression, using to get as shown on the right in Figure 8.
Figure 8 (a) stretches the graph of vertically by a factor of (b) compresses the graph of vertically by a factor of
Stretches and Compressions of the Parent Function
For any factor the function
- is stretched vertically by a factor of if
- is compressed vertically by a factor of if
- has a y-intercept of
- has a horizontal asymptote at a range of and a domain of which are unchanged from the parent function.
Example 4
Graphing the Stretch of an Exponential Function
Sketch a graph of State the domain, range, and asymptote.
- Answer
Before graphing, identify the behavior and key points on the graph.
- Since is between zero and one, the left tail of the graph will increase without bound as decreases, and the right tail will approach the x-axis as increases.
- Since the graph of will be stretched by a factor of
- Create a table of points as shown in Table 4.
Table 4
Draw a smooth curve connecting the points, as shown in Figure 9.
Figure 9
The domain is the range is the horizontal asymptote is
Try It #4
Sketch the graph of State the domain, range, and asymptote.
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 function by we get a reflection about the x-axis. When we multiply the input by we 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 10, and the reflection about the y-axis is shown on the right side of Figure 10.
Figure 10 (a) reflects the graph of about the x-axis. (b) reflects the graph of about the y-axis.
Reflections of the Parent Function
The function
- reflects the parent function about the x-axis.
- has a y-intercept of
- has a range of
- has a horizontal asymptote at and domain of which are unchanged from the parent function.
The function
- reflects the parent function about 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.
Example 5
Writing and Graphing the Reflection of an Exponential Function
Find and graph the equation for a function, that reflects about the x-axis. State its domain, range, and asymptote.
- Answer
Since we want to reflect the parent function about the x-axis, we multiply by to get, Next we create a table of points as in Table 5.
Table 5
Plot the y-intercept, along with two other points. We can use and
Draw a smooth curve connecting the points:
Figure 11
The domain is the range is the horizontal asymptote is
Try It #5
Find and graph the equation for a function, that reflects about the y-axis. State its domain, range, and asymptote.
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 Table 6 to arrive at the general equation for translating exponential functions.
Translations of the Parent Function | |
---|---|
Translation | Form |
Shift
| |
Stretch and Compress
| |
Reflect about the x-axis | |
Reflect about the y-axis | |
General equation for all translations |
Table 6
Translations of Exponential Functions
A translation of an exponential function has the form
Where the parent function, is
- shifted horizontally units to the left.
- stretched vertically by a factor of if
- compressed vertically by a factor of if
- shifted vertically units.
- reflected about the x-axis when
Note the order of the shifts, transformations, and reflections follow the order of operations.
Example 6
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 up units.
- Answer
We want to find an equation of the general form We 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 replace with to get:
- The graph is shifted vertically 4 units, so
Substituting in the general form we get,
The domain is the range is the horizontal asymptote is
Try It #6
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.
Media
Access this online resource for additional instruction and practice with graphing exponential functions.
4.2 Section Exercises
Verbal
1.
What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph?
2.
What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraically?
Algebraic
3.
The graph of is reflected about the y-axis and stretched vertically by a factor of What is the equation of the new function, State its y-intercept, domain, and range.
4.
The graph of is reflected about the y-axis and compressed vertically by a factor of What is the equation of the new function, State its y-intercept, domain, and range.
5.
The graph of is reflected about the x-axis and shifted upward units. What is the equation of the new function, State its y-intercept, domain, and range.
6.
The graph of is shifted right units, stretched vertically by a factor of reflected about the x-axis, and then shifted downward units. What is the equation of the new function, State its y-intercept (to the nearest thousandth), domain, and range.
7.
The graph of is shifted downward units, and then shifted left units, stretched vertically by a factor of and reflected about the x-axis. What is the equation of the new function, State its y-intercept, domain, and range.
Graphical
For the following exercises, graph the function and its reflection about the y-axis on the same axes, and give the y-intercept.
8.
9.
10.
For the following exercises, graph each set of functions on the same axes.
11.
and
12.
and
For the following exercises, match each function with one of the graphs in Figure 12.
Figure 12
13.
14.
15.
16.
17.
18.
For the following exercises, use the graphs shown in Figure 13. All have the form
Figure 13
19.
Which graph has the largest value for
20.
Which graph has the smallest value for
21.
Which graph has the largest value for
22.
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.
23.
24.
25.
For the following exercises, graph the transformation of Give the horizontal asymptote, the domain, and the range.
26.
27.
28.
For the following exercises, describe the end behavior of the graphs of the functions.
29.
30.
31.
For the following exercises, start with the graph of Then write a function that results from the given transformation.
32.
Shift 4 units upward
33.
Shift 3 units downward
34.
Shift 2 units left
35.
Shift 5 units right
36.
Reflect about the x-axis
37.
Reflect about the y-axis
For the following exercises, each graph is a transformation of Write an equation describing the transformation.
38.
39.
40.
For the following exercises, find an exponential equation for the graph.
41.
42.
Numeric
For the following exercises, evaluate the exponential functions for the indicated value of
43.
for
44.
for
45.
for
Technology
For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth.
46.
47.
48.
49.
50.
Extensions
51.
Explore and discuss the graphs of and Then make a conjecture about the relationship between the graphs of the functions and for any real number
52.
Prove the conjecture made in the previous exercise.
53.
Explore and discuss the graphs of and Then make a conjecture about the relationship between the graphs of the functions and for any real number n and real number
54.
Prove the conjecture made in the previous exercise.