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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.**

Explore

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?

Discussion

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

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.

Discussion

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.

$y=a⋅b_{x}$

If the coefficient $a$ is $0,$ the function becomes a horizontal line.
*not* exponential.

$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 If the base $b$ is negative, the function gives undefined results for certain $x-$values. For example, since $b_{1/2}=b ,$ a negative value for $b$ would yield non-real values for $x=21 .$ Hence, a condition on $b$ is needed.

$b≥0 $

However, if $b=0$ or $b=1$, the function becomes a horizontal line.
$y=a⋅0_{x}⇓y=0 and y=a⋅1_{x}⇓y=a $

Therefore, $b$ cannot be equal to $0$ nor $1.$
Pop Quiz

Considering the definitions of an exponential function and a linear function, identify the functions given by the following tables of value.

Example

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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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.

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.

Pop Quiz

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.

Example

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=96000(21 )_{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 $1990?$

a

b See solution.

c $1500$ coyotes

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 $x.$

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.

$y=96000(21 )_{x}$

Substitute

$x=0$

$y=96000(21 )_{0}$

ExponentZero

$a_{0}=1$

$y=96000⋅1$

IdPropMult

Identity Property of Multiplication

$y=96000$

$x$ | $96000(21 )_{x}($ | $y$ |
---|---|---|

$0$ | $96000(21 )_{0}$ | $96000$ |

$1$ | $96000(21 )_{1}$ | $48000$ |

$2$ | $96000(21 )_{2}$ | $24000$ |

$3$ | $96000(21 )_{3}$ | $12000$ |

$4$ | $96000(21 )_{4}$ | $6000$ |

$5$ | $96000(21 )_{5}$ | $3000$ |

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 (0,96000) $

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 $96000$ 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.

$1990−1930=60years $

There are $60$ years from $1930$ to $1990,$ which correspond to $6$ decades. By substituting $6$ for $x$ into the function, the number of kit foxes in $1990$ can be found.
$y=96000(21 )_{x}$

Substitute

$x=6$

$y=96000(21 )_{6}$

PowQuot

$(ba )_{m}=b_{m}a_{m} $

$y=96000⋅2_{6}1_{6} $

BaseOne

$1_{a}=1$

$y=96000⋅2_{6}1 $

CalcPow

Calculate power

$y=96000⋅641 $

MoveLeftFacToNumOne

$a⋅b1 =ba $

$y=6496000 $

CalcQuot

Calculate quotient

$y=1500$

Discussion

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

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.
*expand_more*
*expand_more*
*expand_more*
*expand_more*

$y=10000(0.8)_{x} $

The graph of this function will be drawn as an example by using its function rule.
1

Identify $a$ and $b$

2

Plot the Initial Value

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 $10000,$ so $(0,10000)$ is $y-$intercept of the graph.

3

Use the Constant Multiplier to Find More Points

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 $10000$ and the constant multiplier $0.8.$

$10000⋅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

Lastly, the graph can be drawn by connecting the points with a smooth curve.

Example

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.
### Answer

### Hint

### Solution

$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?

a

b See solution.

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.

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.$Asx→-Asx→+ ∞,∞, y→0y→+∞ $

This piece of information can be illustrated on the graph. Example

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.

**Function Rule:** $y=80(1.25)_{x}$

**Interpretation:** See solution.

Begin by identifying the initial value of the function.

To write an exponential function rule, the initial value of the function $a$ and the constant multiplier $b$ are needed.
The constant multiplier is $b=1.25.$ Therefore, the function rule can be written as follows.

$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.
The point $(1,100)$ lies on the graph. To find the value of $b,$ substitute $x=1$ and $y=100$ into the incomplete rule and solve for $b.$
$y=80⋅b_{x}$

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=80100 $

CalcQuot

Calculate quotient

$b=1.25$

$y=80(1.25)_{x} $

In this context, the initial value means that the initial population when the rabbits were discovered was $80.$ Additionally, the constant multiplier means that each year the population is $1.25$ times more than the previous year. Closure

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 y>2_{x} Boundary Curve 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.

In this case, the scientists can interpret this situation as a disease, for example, that is affecting the rabbits. They can then propose some precautionary measures to maintain a balance in the population of the species.