# Exponential and Logarithmic Functions

## Graphs of Exponential Functions


### Learning Objectives

1. Graph exponential functions (IA 10.2.1).
2. Function transformations (exponential) (CA 3.5.1-3.5.5).

### Objective 1: Graph exponential functions (IA 10.2.1).


### Practice Makes Perfect 


### Objective 2: Function transformations (exponential). (CA 3.5.1-3.5.5)

Vertical and Horizontal Shifts: Given a function  , a new function  where  is a constant, is a vertical shift of the function . All the output values change by k units. If k is a positive, the graph will shift up. If k is negative, the graph will shift down. 
Given a function  , a new function  , where h is a constant, is a horizontal shift of the function  . If h is positive, the graph will shift right. If h is negative, the graph will shift left. 

### Practice Makes Perfect

Function transformations (exponential).
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  change as the input increases by 
 
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
1. the output values are positive for all values of 
 
2. as 
 increases, the output values increase without bound; and
3. as 
 decreases, the output values grow smaller, approaching zero.
 shows the exponential growth function 
 
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  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
1. the output values are positive for all values of 
 
2. as 
 increases, the output values grow smaller, approaching zero; and
3. as 
 decreases, the output values grow without bound.
 shows the exponential decay function, 
 
The domain of 
 is all real numbers, the range is 
 and the 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 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 .
Observe the results of shifting 
 vertically:
1. The domain, 
 remains unchanged.
2. When the function is shifted up 
 units to 




3. When the function is shifted down 
 units to 





### 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 .
Observe the results of shifting 
 horizontally:
1. The domain, 
 remains unchanged.
2. The asymptote, 
 remains unchanged.
3. The y-intercept shifts such that:


### 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 , and the compression, using 
 to get 
 as shown on the right in .

### 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 , and the reflection about the y-axis 
 is shown on the right side of .

### 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  to arrive at the general equation for translating exponential functions.

### Key Equations


### Key Concepts

1. The graph of the function 
 has a y-intercept at 
 domain 
 range 
 and horizontal asymptote 
 See .
2. If 
 the function is increasing. The left tail of the graph will approach the asymptote 
 and the right tail will increase without bound.
3. If 
 the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote 
 
4. The equation 
 represents a vertical shift of the parent function 
 
5. The equation 
 represents a horizontal shift of the parent function 
 See .
6. Approximate solutions of the equation 
 can be found using a graphing calculator. See .
7. The equation 
 where 
 represents a vertical stretch if 
 or compression if 
 of the parent function 
 See .
8. When the parent function 
 is multiplied by 
 the result, 
 is a reflection about the  x-axis. When the input is multiplied by 
 the result, 
 is a reflection about the y-axis. See .
9. All translations of the exponential function can be summarized by the general equation 
 See .
10. Using the general equation 
 we can write the equation of a function given its description. See .

### Section Exercises


### Verbal


### Algebraic 


### Graphical 

For the following exercises, graph the function and its reflection about the y-axis on the same axes, and give the y-intercept.
For the following exercises, graph each set of functions on the same axes.
For the following exercises, match each function with one of the graphs in .
For the following exercises, use the graphs shown in . All have the form 
 
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 of 
 Give the horizontal asymptote, the domain, and the range.
For the following exercises, describe the end behavior of the graphs of the functions.
For the following exercises, start with the graph of 
 Then write a function that results from the given transformation.
For the following exercises, each graph is a transformation of  
  
 Write 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 
 

### Technology 

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

### Extensions 
