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# Graphing Exponential Functions

A function that changes by a constant multiplier is called an exponential function. There are different ways to graph an exponential function — two of them are using a table of values and using the function rule.
Concept

## Exponential Function

Many functions containing a variable exponent, are called exponential functions. Formally, any function that can be written in the following form is an exponential function.

$y=a \cdot b^x$

Here, the coefficient, $a,$ is the $y$-intercept, which is sometimes referred to as the initial value. The base, $b,$ can be interpreted as the constant multiplier. To ensure that $y$ is an exponential function, there are restrictions on $a$ and $b.$

Concept

### $a\neq 0$

If the coefficient $a$ is $0,$ the function becomes a horizontal line. $y=0\cdot b^x \quad \Rightarrow \quad y=0$ This is a line along $y=0,$ and thus, a linear relationship. Therefore, if $a=0$ the function is not exponential.

Concept

### $b \gt 0$ and $b \neq 1$

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 $b$ would yield non-real values for $x=\frac{1}{2}.$ Then, a condition is needed. $b \geq 0$ Furthermore, if $b=0$ or $b=1$, the function becomes a horizontal line. $a\cdot 0^x = 0 \quad \text{and} \quad a\cdot 1^x = a$ Therefore, $b$ can not equal $0$ or $1.$
Therefore, for all exponential functions $y=a \cdot b^x,$ $a\neq 0$ and $b>0, b\neq 1.$
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Exercise

Graph the function $y=3\cdot1.2^x.$

Show Solution
Solution
All functions can be graphed by creating a table of values. To do this, we use arbitrarily chosen $x$-values to find their corresponding $y$-values. We can use this method to graph the given exponential function. Let's start with $x=0.$ Remember to follow the order of operations.
$y=3 \cdot 1.2^x$
$y = 3 \cdot 1.2 ^ {{\color{#0000FF}{0}}}$
$y = 3 \cdot 1$
$y = 3$
Thus, the point $(0,3)$ lies on the given function. We can find other points in the same way. For $x,$ we'll use the whole numbers from $1$ to $5.$
$x$ $3\cdot1.2^x$ $y$
${\color{#0000FF}{1}}$ $3\cdot1.2^{{\color{#0000FF}{1}}}$ $3.6$
${\color{#0000FF}{2}}$ $3\cdot1.2^{{\color{#0000FF}{2}}}$ $\sim4.3$
${\color{#0000FF}{3}}$ $3\cdot1.2^{{\color{#0000FF}{3}}}$ $\sim5.2$
${\color{#0000FF}{4}}$ $3\cdot1.2^{{\color{#0000FF}{4}}}$ $\sim6.2$
${\color{#0000FF}{5}}$ $3\cdot1.2^{{\color{#0000FF}{5}}}$ $\sim7.5$

The points found above all lie on the function. To graph the function, we can plot them in a coordinate plane and connect them with a smooth curve.

Method

## Graphing an Exponential Function using the Function Rule

For an exponential function $y=a \cdot b^x,$ $a$ represents the initial value and $b$ represents the constant multiplier. These values can be used to graph the function. Consider $y=10\,000\cdot 0.8^x.$

### 1

Identify $a$ and $b$

The initial value, $a,$ of an exponential function is the number without an exponent. In this case, $a=10\,000.$ The constant multiplier, $b$ is the number with the exponent. Here, $b=0.8.$

### 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 $10\,000$ so $(0,10\,000)$ is a point on 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 when $x=1$ is $10\,000\cdot 0.8 = 8000.$ Thus, $(1,8000)$ also lies on the graph of the function. Similarly, the point $(2,6400)$ lies on the graph because $8000 \cdot 0.8 = 6400.$ 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.

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Exercise

In $1976,$ scientists discovered a rare population of Flemish Giant rabbits in a secluded forest. Since then, they've been monitoring the population. During the five years of the study, the number of rabbits could be modeled with the exponential function shown.

Use the graph to write the rule for the function, then interpret its initial value and constant multiplier.

Show Solution
Solution

To write an exponential function rule, we need the initial value of the function, $a,$ and the constant multiplier, $b.$ $y=a \cdot b^x$ Notice that the graph starts at $(0,80).$ This means that $80$ is the initial value.

Since $a=80,$ we can write the following incomplete function rule. $y=80\cdot b^x$ To determine $b,$ we can use another point on the graph.

The point $(1,100)$ lies on the graph. Thus, we can susbtitute $x=1$ and $y=100$ into the rule above and solve for $b.$
$y=80 \cdot b^x$
${\color{#009600}{100}}=80\cdot b^{{\color{#0000FF}{1}}}$
Solve for $b$
$100=80\cdot b$
$80\cdot b=100$
$b=\dfrac{100}{80}$
$b=1.25$
The constant multiplier is $b=1.25.$ Thus, the function rule can be written as follows. $y=80\cdot 1.25^x$ Next, we can interpret the values of $a$ and $b$ we found above. The initial value, $a=80$, means that the initial population when the rabbits were discovered was $80.$ Additionally, a constant multiplier of $1.25$ means that each year the population is $1.25$ times more than the previous year.
Method

## Solving Exponential Equations Graphically

If the dependent variable of an exponential function written in the form $y = a \cdot b^x,$ is exchanged for a constant, say $C,$ the result is a one-variable equation: $C = a \cdot b^x.$

This type of equation is called an exponential equation, and can be solved graphically. This is done by first graphing the function $y = a \cdot b^x,$ then finding the $x$-coordinate of the point(s) on the graph with the $y$-coordinate $C.$ The $x$-coordinate(s) is the solution to the equation.
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Exercise

Use the graph to solve the equation $3 = 5 \cdot 0.85^x.$

Show Solution
Solution

The graph shows all $x$-$y$ points that satisfy the function rule $y = 5 \cdot 0.85^x.$ Let's compare the function rule and the equation. \begin{aligned} \textbf{Function rule:} \quad y = 5 \cdot 0.85^x\\ \textbf{Equation:} \quad 3 = 5 \cdot 0.85^x \end{aligned} The only difference between these two equalities is that the independent variable, $y,$ is replaced by a $3$ in the equation. Thus, we solve the equation by finding the $x$-coordinate of any point on the graph that has the $y$-coordinate $3.$

We can identify one such point in the graph. Let's now find the $x$-coordinate of this point graphically.

This $x$-coordinate is not easily read from the graph, so we'll have to make an approximation. It's just a bit bigger than $3,$ so we'll use $3.1.$ This means that an approximate solution to the equation is $x \approx 3.1.$ We can verify this by substituting it into equation to see if a true statement is made.

$3 = 5 \cdot 0.85^x$
$x \approx {\color{#0000FF}{3.1}}$
$3 \overset{?}{\approx} 5 \cdot 0.85^{{\color{#0000FF}{3.1}}}$
$3 \approx 3.02112 \ldots$

The right-hand side and the left-hand side are approximately equal, so we have indeed found an approximate solution to the equation: $x \approx 3.1.$