3. Graphing Exponential Functions
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Chapter 6
3.
Graphing Exponential Functions
Exponential functions are powerful mathematical tools that model various real-life situations, from population growth to financial investments. Unlike linear functions, exponential functions have a dependent variable multiplied by a constant factor as its independent variable changes by a set amount. The initial value, often referred to as the y-intercept, and the constant multiplier are crucial components of these functions. By understanding and graphing these functions, one can interpret and predict patterns, such as population trends or investment growth. For instance, scientists might use exponential functions to model the growth of a particular animal species over time. By analyzing the graph, they can make informed decisions and predictions about future population sizes.
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Lesson Settings & Tools
| | 12 Theory slides |
| | 9 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Graphing Exponential Functions
Slide of 12
Earlier in this course, one of the function families called linear functions was covered. However, linear functions are not always the best option for modeling real-life situations. For example, another function family is needed to model populations, carbon date artifacts, compute investments. This lesson will introduce this new function family.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Investigating Function Families
Each function family has unique characteristics. By investigating the table of values of a function, several characteristics can be identified. The following tables of values belong to two functions derived from different function families.
How do the y-values of Function I and Function II change according to their corresponding x-values? What do the graph of these functions look like? Which function family does Function I belong? What can be concluded about Function II?
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The Concept of Exponential Expressions
To understand the functions whose dependent variable is multiplied by a constant factor as its independent variable changes by a constant amount, one algebraic expression needs to be mentioned beforehand.
Concept
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Exponential Expression
An exponential expression is a type of algebraic expression which consists of a number, called the base, raised to either a number, a variable, or an expression, called the exponent. Sometimes an exponential expression may be multiplied by another number, called a coefficient.
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Exponential Function
An exponential function is a nonlinear function that can be written in the following form, where a≠ 0, b>0, and b≠ 1. As the independent variable x changes by a constant amount, the dependent variable y multiplied by a constant factor. Therefore, consecutive y-values form a constant ratio.
y=a * b^x
Why
Why a≠ 0?If the coefficient a is 0, the function becomes a horizontal line.
y=0* b^x ⇒ y=0
This is a line along the x-axis and, therefore, is a linear relation. This means that if a=0, then the function is not exponential.
Why
Why b> 0 and b≠ 1?If the base b is negative, the function gives undefined results for certain x-values. For example, since b^(1/2)=sqrt(b), a negative value for b would yield non-real values for x= 12. Hence, a condition on b is needed.
b ≥ 0
However, if b= 0 or b= 1, the function becomes a horizontal line.
ccc
y=a* 0^x & & y=a* 1^x ⇓ & and& ⇓ y=0 & & y=a
Therefore, b cannot be equal to 0 nor 1.
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Identifying Functions
Considering the definitions of an exponential function and a linear function, identify the functions given by the following tables of value.
Example
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Population of Coyotes
Coyotes tend to thrive along the coasts, the deserts, and forests of the United States. A case study has shown that since 1950 the population y of coyotes in a particular national park triples every 20 years.
This population growth of coyotes can be modeled by an exponential function. y=18(3)^x In this function, x is the number of 20-year periods. Using the function of coyote population growth, answer the following questions.
a How many coyotes were in the national park in 1950?
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b How many coyotes will be in the national park after 40 more years?
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Hint
a What value of x should be used to find the number of coyotes in 1950?
b Identify the value of x corresponding to a 40-year period.
Solution
a Recall that x is the number of 20-year periods. In 1950, the value of x was 0. Therefore, by substituting 0 for x into the function, the initial number of coyotes can be found.
y=18(3)^x
Substitute
x= 0
y=18(3)^0
ExponentZero
a^0=1
y=18* 1
IdPropMult
Identity Property of Multiplication
y=18
b Since x is the number of 20-year periods, 40 years correspond to two periods. With this information, the number of coyotes in the national park can be found as follows.
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Evaluating Exponential Functions
The following applet provides several exponential functions. Evaluate the value of the function for the given value of x. If it is necessary, round the result to two decimal places.
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Population of Foxes
The San Joaquin kit fox was relatively common until the 1930s, when people began converting grasslands to farms, orchards, and cities.
Since then, the number y of the San Joaquin kit foxes has been decreasing by 50 % every decade. The following exponential function shows the number of foxes since the year 1930. y=96 000(1/2)^x In this function, x is the number of decades. Using the function, answer the following questions.
a Graph the function.
b Find and interpret the y-intercept.
c How many kit foxes were there in 1990?
Answer
a 
b See solution.
c 1500 coyotes
Hint
a Make a table of values to graph the function.
b The y-intercept is a point where the graph intersects the y-axis.
x Begin by determining the value of x.
Solution
a All functions can be graphed by creating a table of values. To do this, arbitrarily chosen x-values are used to find their corresponding y-values. This method can be used to graph the given exponential function. Begin by finding the y-value corresponding to x=0. Remember to follow the order of operations.
y=96 000(1/2)^x
Substitute
x= 0
y=96 000(1/2)^0
ExponentZero
a^0=1
y=96 000* 1
IdPropMult
Identity Property of Multiplication
y=96 000
| x | 96 000(1/2)^x | y |
|---|---|---|
| 0 | 96 000(1/2)^0 | 96 000 |
| 1 | 96 000(1/2)^1 | 48 000 |
| 2 | 96 000(1/2)^2 | 24 000 |
| 3 | 96 000(1/2)^3 | 12 000 |
| 4 | 96 000(1/2)^4 | 6000 |
| 5 | 96 000(1/2)^5 | 3000 |
The points found above all lie on the function. To graph the function, plot them in a coordinate plane and connect them with a smooth curve. Note that the number of decades cannot be negative, so the function will be restricted by the first quadrant.
b Considering the table and graph from Part A, the y-intercept can be identified. It is a point where the curve intersects the y-axis. Hence, the x-coordinate is 0.
y-intercept ( 0,96 000) The x-coordinate of the point represents the number of decades since 1930. The y-coordinate of the point represents the number of kit foxes. Since the function shows the population starting from 1930, the y-intercept tells that there were 96 000 kit foxes in 1930.
c To find the number of kit foxes in 1990, first the number of decades since 1930 needs to be determined.
y=96 000(1/2)^x
Substitute
x= 6
y=96 000(1/2)^6
PowQuot
(a/b)^m=a^m/b^m
y=96 000* 1^6/2^6
BaseOne
1^a=1
y=96 000* 1/2^6
CalcPow
Calculate power
y=96 000* 1/64
MoveLeftFacToNumOne
a* 1/b= a/b
y=96 000/64
CalcQuot
Calculate quotient
y=1500
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Graphing an Exponential Function
The most basic method for graphing a function is making a table of values. This method can be used to graph an exponential function, as well. On the other hand, there is another method to graph an exponential function more effectively.
Method
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Graphing an Exponential Function Using the Function Rule
For an exponential function y=a * b^x, a represents the initial value and b represents the constant multiplier. These values can be used to graph the function. Consider the following exponential function.
y=10 000(0.8)^x
The graph of this function will be drawn as an example by using its function rule.
1
Identify a and b
expand_more 2
Plot the Initial Value
expand_more The initial value is the y-value when x=0. It can also be thought of as the y-intercept of the function. Here, the initial value is 10 000, so (0,10 000) is y-intercept of the graph.
3
Use the Constant Multiplier to Find More Points
expand_more When the x-value increases by 1, the y-value is multiplied by b. Since b=0.8, the y-value for x=1 can be calculated as the product of the initial value 10 000 and the constant multiplier 0.8. 10 000* 0.8 = 8000 Therefore, (1,8000) also lies on the graph of the function. Similarly, the point (2,6400) lies on the graph because 8000 * 0.8 = 6400. These points are shown on the graph.
This process can be repeated until a general form of the graph emerges.
4
Draw the Curve
expand_more Lastly, the graph can be drawn by connecting the points with a smooth curve.
Example
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Graphing an Exponential Function and Identifying Its Key Features
The graph of an exponential function can be drawn by using its function rule. Then from the graph of a function the key features such as domain, range, and end behavior can be found. y=3(2)^x Considering the above function, answer the following questions.
a Graph the function using its function rule.
b What are the key features of the function?
Answer
a 
b See solution.
Hint
a Begin by identifying the initial value value and constant multiplier of the function.
b Use the graph of the function to find the domain, range, and end behavior of the function. Recall the definitions of these concepts.
Solution
a The function y=3(2)^x has the initial value a=3 and the constant multiplier b= 2. Use these values to mark four points on the function's graph.
The function can now be graphed by connecting the points with a smooth curve.
b The function's key features can be described by interpreting its graph.
- The graph has a y-intercept at (0,3).
- The function y=3(2)^x is greater than 0 for all x. Although the left-end of the graph approaches the x-axis, it never intersects it. Therefore, there is no x-intercept.
- As x approaches ∞, the function also approaches ∞. Therefore, the function increases for all x.
Combining these features, it can be also concluded that the domain of the function is all real numbers and its range is positive real numbers.
Looking at the graph, it can be seen that the left-end approaches y=0 and the right-end extends upward. With this information, the end behavior of y=3(2)^x can be written as follows. As x → - &∞, && y → 0 As x → + &∞ , && y → +∞ This piece of information can be illustrated on the graph.
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Population of Flemish Giant Rabbits
By applying reverse engineering on the graph of an exponential function, its function rule can be written as well. In 1976, scientists discovered a rare population of Flemish Giant rabbits in a secluded forest.
Since then, they have been monitoring the population. After five years of conducting the study, the number of rabbits could be modeled with the following exponential function.
Use the graph to write the rule for the function. Then, interpret its initial value and constant multiplier.
Answer
Function Rule: y=80(1.25)^x
Interpretation: See solution.
Hint
Begin by identifying the initial value of the function.
Solution
To write an exponential function rule, the initial value of the function a and the constant multiplier b are needed. y=a(b)^x Notice that the graph starts at (0,80). This means that 80 is the initial value.
Since a=80, the following incomplete function rule can be written. y=80(b)^x To determine b, use another point on the graph.
y=80 * b^x
SubstituteII
x= 1, y= 100
100=80(b)^1
▼
Solve for b
ExponentOne
a^1=a
100=80* b
RearrangeEqn
Rearrange equation
80* b=100
DivEqn
.LHS /80.=.RHS /80.
b=100/80
CalcQuot
Calculate quotient
b=1.25
Closure
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Exponential Inequalities
Throughout the lesson, evaluating and graphing an exponential function and interpreting the graph of an exponential function have been covered. Beside these, exponential inequalities can be also mentioned even though they are not frequently used. y > a(b)^x y < a(b)^x y ≥ a(b)^x y ≤ a(b)^x They are useful in situations involving repeated multiplication, especially when being compared to a constant value, such as in the case of interest. Their graphs can be drawn similarly to graphs of linear inequalities. For example, consider the following exponential inequality. y > 2^x Notice that the y-variable is already isolated. Since it is an exponential inequality, there will be a boundary curve instead of a boundary line. Inequality & Boundary Curve y > 2^x & y = 2^x The curve can be graphed using the function rule of the exponential function. Note that the inequality is strict, so the curve will be dashed.
Finally, since all of the y-values are greater than 2^x, the region above the curve will be shaded.
Consider the example from the collection where scientists modeled the number of Flemish Giant rabbits using y=80(1.25)^x. They are expecting the number of rabbits to increase along with the curve formed by the exponential function. It can be assumed that the number of rabbits falls below the curve at any given period of time.
Graphing Exponential Functions
Exercise 2.1
Determine whether each of the following statements is always, sometimes, or never true.
|
An exponential function whose domain is the set of all real numbers has a y-intercept. |
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|
When modeling a real-life situation with an exponential function, the domain is the set of all real numbers. |
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Let's begin by reviewing the general form of an exponential function. f(x)=ab^x In this form, a≠ 0 is the y-intercept or initial value of the function. This means that, if the domain of the function is the set of all real numbers, the function will always pass through the point (0,a).
Therefore, exponential functions always have a y-intercept.
Without any context, an exponential function can be defined over all real numbers. However, when modeling a real-life situation, sometimes the domain must be defined for only non-negative numbers. For example, if the independent variable t represents time, it does not make sense that t takes negative values.
This why, when modeling this type of situations, exponential functions can be defined for non-negative values. Depending on the context of the problem the independent variable can take different types of values. Therefore, the answer is sometimes.
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When Ignacio was born, his parents deposited money in an account with a fixed interest rate. The graph below shows how this amount increases over time.
Here, x represents the number of years that have passed since the account was created, and y represents the amount of dollars in the account.
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The y-intercept of an exponential function is also known as the initial value. It is found by looking at the y-value when x=0. Let's find the y-intercept of the given graph.
The y-intercept occurs when y=5000. This means that Ignacio's parents initially deposited $5000.
We want to find how much money there will in the account after 20 years. We can use the graph to estimate this value. To do this, we will find the y-value when x=20.
We can see that when x= 20 the value of y is 10 000. Therefore, after 20 years, there will be $10 000.
Let's take another look at the graph. Remember that the y-values represent the amount of dollars in the account and that the x-values represent the number of years that have passed since the account was created.
We can notice that the function is defined in the first quadrant. In this quadrant the values of x are positive, but we can also notice that the function has a y-intercept, so it includes x=0 in its domain. Domain: x ≥ 0 This means that the domain of the function is all non-negative numbers. Having negative x-values does not makes sense because it would mean that the account already existed before it was created. This would be like going back in time!
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Graphing Exponential Functions
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