Exponential Functions: Properties, Graphs, and Applications

Exponential Functions

Definition and Basic Properties

An exponential function is a function of the form $f(x) = Ca^x$, where $a$ is a positive real number ($a > 0$ and $a \neq 1$), $C \neq 0$ is a real number, and $x$ is any real number. The domain of $f$ is all real numbers, and the base $a$ is called the growth (or decay) factor. The constant $C$ is the initial value since $f(0) = C$.

• Domain: $(-\infty, \infty)$ (all real numbers)

• Range: $(0, \infty)$ (all positive real numbers if $C > 0$)

• y-intercept: $(0, C)$

• x-intercept: None

• Horizontal asymptote: $y = 0$

Growth vs. Decay:

• If $a > 1$, the function models exponential growth.

• If $0 < a < 1$, the function models exponential decay.

Key Points on the Graph: The graph of $f(x) = a^x$ always passes through $(0, 1)$, $(1, a)$, and $(-1, 1/a)$.

Laws of Exponents

The following properties hold for all real numbers $s, t, a, b$ with $a > 0$ and $b > 0$:

• $a^0 = 1$

• $a^s \cdot a^t = a^{s+t}$

• $(a^s)^t = a^{st}$

• $a^{-s} = \frac{1}{a^s}$

• $\frac{a^s}{a^t} = a^{s-t}$

• $(ab)^s = a^s b^s$

Evaluating Exponential Expressions

To evaluate $a^x$ using a calculator:

1. Enter the base $a$.

2. Press the $y^x$ key (or caret ^).

3. Enter the exponent $x$.

4. Press = or ENTER.

Example: Evaluate $5^{1.7}$, $5^{1.73}$, $5^{1.732}$, $5^{1.73205}$, $5^3$ using a calculator.

Identifying Linear vs. Exponential Functions

Comparing Patterns in Data

To distinguish between linear and exponential functions, examine how the output values change:

• Linear function: The difference between consecutive outputs is constant.

• Exponential function: The ratio of consecutive outputs is constant.

Example: For $f(x) = 2^x$ and $g(x) = 2x + 3$, evaluate at $x = -2, -1, 0, 1, 2, 3$ and observe the patterns.

Table: Average Rate of Change vs. Ratio of Consecutive Outputs

x	f(x) = y	Average Rate of Change	Ratio of Consecutive Outputs
-1	-4.5
0	-3
1	-1.5
2	0
3	1.5

Additional info: Similar tables can be constructed for other functions to compare linear and exponential behavior.

Properties of the Exponential Function

Summary of Key Properties

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$

• x-intercepts: None

• y-intercept: $(0, 1)$

• Horizontal asymptote: $y = 0$ (the x-axis)

• If $0 < a < 1$, the function is decreasing (decay).

• If $a > 1$, the function is increasing (growth).

• The graph is smooth and continuous with no corners or gaps.

• Contains the points $(-1, 1/a)$, $(0, 1)$, $(1, a)$.

The Number e and Continuous Growth

Definition of e

The number $e$ is a mathematical constant that arises as the limit of $(1 + 1/n)^n$ as $n$ approaches infinity:

$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$

As the number of compounding periods increases, the growth factor approaches $e$.

Table: Approximating e

n	$(1 + 1/n)^n$
10	2.5937
50	2.6916
100	2.7048
500	2.7169
1000	2.7169
10,000	2.7181
100,000	2.7183
1,000,000	2.7183

Additional info: The value of $e$ is approximately 2.7183.

Evaluating Exponential Expressions with Base e

• To evaluate $e^x$ on a calculator, use the $e^x$ key.

• Examples:

  • $e^4 \approx 54.5982$

  • $e^{-0.57} \approx 0.5650$

  • $e^0 = 1$

Graphing Exponential Functions and Transformations

To graph functions such as $f(x) = 2 \cdot 3^{x+1} - 4$, apply transformations to the basic exponential graph:

• Horizontal shift: $x + 1$ shifts left by 1 unit.

• Vertical stretch: Multiply by 2.

• Vertical shift: Subtract 4 (down 4 units).

Domain: $(-\infty, \infty)$ Range: $(-4, \infty)$ Horizontal asymptote: $y = -4$

Graph of $f(x) = e^x$ and Transformations

The graph of $f(x) = e^x$ passes through $(-1, 1/e)$, $(0, 1)$, $(1, e)$, $(2, e^2)$, etc. Transformations can shift, stretch, or reflect the graph.

Solving Exponential Equations

Equating Bases

To solve equations like $a^u = a^v$, set the exponents equal: $u = v$ (provided $a > 0$, $a \neq 1$).

Example: Solve $5^{x+3} = 125$ by rewriting $125$as $5^3$, then set $x+3 = 3$.

For more complex equations, manipulate both sides to have the same base, then equate exponents.

Solving Graphically

Set $Y_1$ equal to the left side and $Y_2$ to the right side in a graphing utility. The intersection gives the solution.

Applications of Exponential Functions

Probability Example: Car Arrivals

Suppose cars arrive at a rate of 6 per hour (0.1 per minute). The probability that a car arrives within $t$ minutes is:

$F(t) = 1 - e^{-0.1t}$

• (a) For $t = 10$, $F(10) = 1 - e^{-1} \approx 0.6321$

• (b) For $t = 40$, $F(40) = 1 - e^{-4} \approx 0.9817$

• (c) As $t \to \infty$, $F(t) \to 1$

• (d) The graph of $F$ is an increasing curve approaching 1 as $t$ increases.

• (e) To find when $F(t) = 0.5$, solve $1 - e^{-0.1t} = 0.5$; $t = -10 \ln(0.5) \approx 6.93$ minutes.