Exponential Growth

In Chapter 2, we looked at linear growth where there is a constant rate of change. Unlike linear growth that increases by adding a constant value to [latex]y[/latex] for every unit increase in [latex]x[/latex], exponential growth increases by multiplying by a constant that is neither equal to 0 nor 1. Table 1 shows exponential growth as each [latex]y[/latex]-value increases by a multiple of 2. Notice that the values in the [latex]x[/latex] column correspond to the exponents in the fourth column. This is because the values in the [latex]x[/latex] column record the number of times we have multiplied by 2.

[latex]x[/latex]	[latex]y[/latex]
1	2	[latex]= 2[/latex]	[latex] =2^1[/latex]
2	4	[latex]= 2 \cdot 2[/latex]	[latex] =2^2[/latex]
3	8	[latex]= 2 \cdot 2 \cdot 2[/latex]	[latex] =2^3[/latex]
4	16	[latex]=2 \cdot 2 \cdot 2 \cdot 2[/latex]	[latex] =2^4[/latex]
5	32	[latex]=2 \cdot 2 \cdot 2 \cdot 2 \cdot 2[/latex]	[latex] =2^5[/latex]
Table 1. Exponential growth

The pattern shown in the fourth column of table 1 indicates that exponential growth is repeated multiplication by a factor of 2 so the equation for the pattern is [latex]y= 2^x[/latex].

Zero and Negative Exponents

Table 1 shows the pattern of exponential growth of [latex]2^x[/latex] when [latex]x[/latex] is a natural number. But what is the value of [latex]2^x[/latex] when the exponent [latex]x[/latex] is zero or a negative number? If we consider the pattern in Table 1 in the reverse direction we can find the value of [latex]y=2^x[/latex] when [latex]x=0, -1, -2,…[/latex]. As the exponent decreases by 1, the value of [latex]y=2^x[/latex] is divided by 2.

For example, [latex]2^x=8[/latex] when the exponent [latex]x[/latex] is 3 (i.e., [latex]y=2^3=8[/latex]). As [latex]x[/latex] decreases by 1 to 2, the [latex]y[/latex] value is 4 (See Table 2). In other words, 8 is divided by 2 to get 4. Following this logic, the [latex]y[/latex] value will be 4 ÷ 2 = 2 when [latex]x[/latex] decreases by 1 again to 1. Following this pattern, the [latex]y[/latex] value will be 2 ÷ 2 = 1 when [latex]x[/latex] decreases by 1 again to zero. Therefore, [latex]2^0=1[/latex]. In fact,  [latex]a^0=1[/latex] for all values of [latex]a[/latex], [latex]a\ne 0[/latex]. [latex]0^0[/latex] is undefined.

Continuing this reverse pattern, the value of [latex]y[/latex] will be [latex]1\div 2=\dfrac{1}{2}[/latex] when [latex]x=-1[/latex]. Then, dividing by 2 again gives [latex]y=\dfrac{1}{2}÷2=\dfrac{1}{2}\cdot\dfrac{1}{2}=\dfrac{1}{4}[/latex] when [latex]x=-2[/latex]. Recall that dividing by 2 is equivalent to multiplying by [latex]\dfrac{1}{2}[/latex]. This pattern shows that the value of [latex]y[/latex] is a fraction where [latex]y=\dfrac{1}{2^{|x|}}[/latex] when the exponent [latex]x[/latex] is a negative number (See Table 2). For example, [latex]y=2^{-3}=\dfrac{1}{2^{|-3|}}=\dfrac{1}{2^3}[/latex]. In fact, the value of [latex]y=a^x[/latex] will alway be [latex]\dfrac{1}{a^{|x|}}[/latex] when the exponent [latex]x[/latex] is a negative number and the base [latex]a ≥ 0[/latex].

Table 3 shows how the values of [latex]y=2^x[/latex] for the exponents [latex]x=0, -1, -2, -3…[/latex] are obtained following the reverse pattern starting at [latex]x=0[/latex].

[latex]x[/latex]	[latex]y[/latex]	Method for Obtaining [latex]y[/latex]	Equation
-3	[latex]\dfrac{1}{8}[/latex]	[latex]=\dfrac{1}{2} \cdot \dfrac{1}{2} \cdot \dfrac{1}{2}[/latex]	[latex]2^{-3}[/latex]
-2	[latex]\dfrac{1}{4}[/latex]	[latex]=\dfrac{1}{2} \cdot \dfrac{1}{2}[/latex]	[latex]2^{-2}[/latex]
-1	[latex]\dfrac{1}{2}[/latex]	[latex]=\dfrac{1}{2}[/latex]	[latex]2^{-1}[/latex]
0	1	[latex]= 2 \cdot \dfrac{1}{2}[/latex] = 1	[latex] 2^0[/latex]
1	2	[latex]=2[/latex]	[latex] 2^1[/latex]
2	4	[latex]= 2 \cdot 2[/latex]	[latex] 2^2[/latex]
3	8	[latex]=2 \cdot 2 \cdot 2[/latex]	[latex] 2^3[/latex]
4	16	[latex]=2 \cdot 2 \cdot 2 \cdot 2[/latex]	[latex] 2^4[/latex]
5	32	[latex]=2 \cdot 2 \cdot 2 \cdot 2 \cdot 2[/latex]	[latex] 2^5[/latex]
Table 2. Exponential growth with base 2

Zero and negative exponents will be discussed again in section 5.3.2 using the Product and Quotient Rule for Exponents.

The Difference between Exponential Growth and Power Growth

In exponential growth, the variable is the exponent (e.g., [latex]2^x[/latex]). In power growth, the variable is the base (e.g., [latex]x^2[/latex]). Exponential growth grows faster than power growth. For example, table 3 shows that the exponential growth [latex]2^x[/latex] grows much faster than the power growth [latex]x^2[/latex] while table 4 shows that the exponential growth [latex]3^x[/latex] grows much faster than the power growth [latex]x^3[/latex].

[latex]x[/latex]	[latex]x^2[/latex]	[latex]2^x[/latex]		[latex]x[/latex]	[latex]x^3[/latex]	[latex]3^x[/latex]
0	0	1		0	0	1
1	1	2		1	1	3
2	4	4		2	8	9
3	9	8		3	27	27
4	16	16		4	64	81
5	25	32		5	125	243
6	36	64		6	216	729
7	49	128		7	343	2187
8	64	256		8	512	6561
Table 3. Exponential versus power growth				Table 4. Exponential versus power growth

Initial Value

The initial value of exponential growth occurs at [latex]x=0[/latex]. So far we have considered only initial values of 1. Tables 1 and 2 show a pattern where [latex]y=1\cdot 2^x[/latex]. However, the initial value can be any real number.

For example, let’s consider an initial value of 8 and an exponential growth rate of 2. We can create a table that illustrates this scenario starting with [latex]y=8[/latex] when [latex]x=0[/latex], then multiplying each [latex]y[/latex]-value by 2:

[latex]x[/latex]	[latex]y[/latex]	Equation
-2	[latex]4\div 2=2[/latex]	[latex]8\cdot2^{-2}[/latex]
-1	[latex]8\div 2=4[/latex]	[latex]8\cdot2^{-1}[/latex]
0	[latex]8[/latex]	[latex]y=8\cdot2^0[/latex]
1	[latex]8\cdot 2=16[/latex]	[latex]8\cdot2^1[/latex]
2	[latex]16\cdot 2=32[/latex]	[latex]8\cdot2^2[/latex]
3	[latex]32\cdot 2=64[/latex]	[latex]8\cdot2^3[/latex]

The equation that models an exponential growth rate of 2 with an initial value of 8 is [latex]y=8\cdot2^x[/latex].

Exponential Functions and Their Graphs

Exponential growth has an initial value and an exponential rate of change. The initial value occurs at [latex]x=0[/latex]. In table 1, the initial value is 1 (when [latex]x=0[/latex]), and the exponential rate of change is 2. This creates a pattern where [latex]y=1\cdot 2^x[/latex]. Consequently, the exponential growth in table 1 may be modeled or represented by the function [latex]f(x) = 2^x[/latex].

If we graph the values [latex](x, y)[/latex] from table 1, we can then connect the points to draw the graph the exponential function [latex]f(x)=2^x[/latex] (figure 1).

Figure 2 illustrates how the graph changes as the value of [latex]r[/latex] changes. Move the red circle up or down to change the value of [latex]r[/latex] and watch what happens to the function.

Figure 2. Interactive graph [latex]f(x)=r^x[/latex]

Manipulate the graph of [latex]f(x)=r^x[/latex] in figure 2 to answer the following questions:

1. What happens to the point (0, 1) as [latex]r[/latex]changes?

The point (0, 1) never changes. The point (0, 1) is always on the graph of [latex]f(x)=r^x[/latex].

2. What happens to the point on the graph at [latex]x=1[/latex] as [latex]r[/latex]changes?

At [latex]x=1[/latex], the point on the graph will always be [latex](1, r)[/latex], because [latex]f(1)=r^1=r[/latex].

3. What happens to the graph when [latex]r=1[/latex]?

If [latex]r=1[/latex] we get a flat line; a linear equation [latex]y=1[/latex]. This is why [latex]r[/latex] is never allowed to equal 1.

4. What happens to the graph when [latex]r=0[/latex]?

If [latex]r=0[/latex] we get a flat line starting at [latex]x>0[/latex]. [latex]f(0)=0^0[/latex] which is undefined.  For any [latex]x<0[/latex], [latex]0^{\text{negative number}}=\dfrac{1}{0^{\text{positive number}}}=\dfrac{1}{0}[/latex], which is undefined. This is why [latex]r[/latex] is never allowed to equal 0.

5. What happens to the graph when [latex]r<0[/latex]?

When [latex]r<0[/latex], the graph disappears!! This is why [latex]r[/latex] is a positive real number ≠ 0, 1.

6. What happens to the graph when [latex]r>1[/latex]?

The graph comes up from [latex]y=0[/latex], passes through (0, 1) and (1, r), then quickly moves towards [latex]+\infty[/latex].

7. What happens to the graph when [latex]0<r<1[/latex]?

When [latex]r[/latex] is a proper fraction, the graph comes in from [latex]+\infty[/latex], passes through (0, 1) and (1, r), then moves slowly towards [latex]y=0[/latex].

The Asymptote and Intercepts

A significant feature on the graph of any exponential function [latex]f(x) = r^x[/latex] ([latex]r>0,\;r\neq1[/latex]) is that the graph never crosses the [latex]x[/latex]-axis. It continually approaches the [latex]x[/latex], getting closer and closer, but the graph never meets the [latex]x[/latex]-axis. Figure 2 illustrates that when [latex]0<r<1,[/latex] the graph gets closer and closer to the [latex]x[/latex]-axis when [latex]x[/latex] gets closer and closer to positive infinity (figure 3). However, when [latex]r>1[/latex], the graph gets closer and closer to the [latex]x[/latex]-axis when [latex]x[/latex] gets closer and closer to negative infinity (figure 4).

Figure 3. [latex]f(x)=r^x[/latex] with [latex]0<r<1[/latex]	Figure 4. [latex]f(x)=r^x[/latex] with [latex]r>1[/latex]

Why does the graph never meet the [latex]x[/latex]-axis? Consider the following examples using the function [latex]f(x)=2^x[/latex] that was graphed in figure 1:

[latex]f(-10)=2^{-10}=\dfrac{1}{2^{10}}[/latex]

[latex]f(-100)=2^{-100}=\dfrac{1}{2^{100}}[/latex]

[latex]f(-100000)=2^{-100000}=\dfrac{1}{2^{100000}}[/latex]

As the the value of [latex]x[/latex] gets closer to negative infinity, the value of the function [latex]y[/latex] is a fraction with a numerator of 1 and a denominator that is a very large positive number. The value of [latex]x[/latex] gets more and more negative as it gets closer to negative infinity, so the value of the function will get smaller and smaller. It will get close to zero but will never be zero because [latex]\dfrac{1}{\text{very large positive number}}[/latex] is always positive and therefore greater than zero.

Figure 2 shows that for all values of [latex]r>0[/latex] and [latex]r\neq1[/latex], the graph gets close to but never crosses the [latex]x[/latex]-axis, it is a horizontal asymptote of the function [latex]f(x)=r^x[/latex]. Also, since the graph never meets the [latex]x[/latex]-axis, there is no [latex]x[/latex]-intercept for the function. The [latex]y[/latex]-intercept of the function [latex]f(x)=r^x[/latex] is always (0, 1).

We use a dotted line to show that a graph has a horizontal asymptote (figure 5).

Domain and Range

Figure 5 shows the graph of [latex]f(x)=2^x[/latex]. The domain of the function is the set of all possible [latex]x[/latex]-values, so domain = [latex]\{x\;|\;x\in \mathbb{R}\}[/latex]. Any [latex]x[/latex]-value from [latex]-\infty[/latex] to [latex]+\infty[/latex] has a corresponding function value. It’s range, the set of all function values, lies above the line [latex]y=0[/latex]. Consequently, the range =[latex]\{f(x)\;|\;f(x)\in\mathbb{R}^+\}[/latex], where [latex]\mathbb{R}^+[/latex] is the set of all positive real numbers.

The exponential function [latex]f(x)=r^x[/latex] is the parent function of all exponential functions. In the next section, we will see what happens to the graph of the function when we transform the parent function.