Exponential Functions and Their Graphs

Exponential Functions and Graphs

Introduction

Exponential functions are a fundamental class of functions in algebra, characterized by a constant base raised to a variable exponent. They are widely used in modeling growth and decay in natural and applied sciences. This section covers the definition, properties, graphing techniques, and applications of exponential functions.

Exponential Functions

Definition and Properties

• Exponential Function: An exponential function is defined as $f(x) = a^x$, where $x$ is a real number, $a > 0$, and $a \neq 1$.

• Base: The constant $a$ is called the base of the exponential function. The base must be positive to avoid complex numbers when taking even roots of negative numbers.

• Examples:

  • $f(x) = 2^x$

  • $f(x) = \left(\frac{1}{2}\right)^x$

  • $f(x) = (3.57)^x$

Graphing Exponential Functions

Steps for Graphing

• 1. Compute several function values and organize them in a table.

• 2. Plot the points on a coordinate plane and connect them with a smooth curve. Ensure enough points are plotted to capture the curve's steepness.

Example: Graphing $y = f(x) = 2^x$

x	y = f(x) = 2^x	(x, y)
0	1	(0, 1)
1	2	(1, 2)
2	4	(2, 4)
3	8	(3, 8)
-1	0.5	(-1, 0.5)
-2	0.25	(-2, 0.25)
-3	0.125	(-3, 0.125)

• As $x$ increases, $y$ increases without bound.

• As $x$ decreases, $y$ approaches 0 but never reaches it.

• The x-axis ($y = 0$) is a horizontal asymptote.

Example: Graphing $y = f(x) = \left(\frac{1}{2}\right)^x$

• This function is equivalent to $y = 2^{-x}$, which is a reflection of $y = 2^x$ across the y-axis.

x	Points of $f(x) = 2^x$	Points of $f(x) = (1/2)^x$
0	1	1
1	2	0.5
2	4	0.25
3	8	0.125
-1	0.5	2
-2	0.25	4
-3	0.125	8

• As $x$ increases, $y$ approaches 0.

• As $x$ decreases, $y$ increases without bound.

• The x-axis ($y = 0$) remains a horizontal asymptote.

Comparing Graphs of Exponential Functions

• For $0 < a < 1$, the function $f(x) = a^x$ decreases as $x$ increases (decay).

• For $a > 1$, the function $f(x) = a^x$ increases as $x$ increases (growth).

• All exponential functions of the form $f(x) = a^x$ have the y-intercept at $(0, 1)$.

Transformations of Exponential Functions

• Horizontal Shifts: $f(x) = a^{x-h}$ shifts the graph right by $h$ units.

• Vertical Shifts: $f(x) = a^x + k$ shifts the graph up by $k$ units.

• Reflections: $f(x) = a^{-x}$ reflects the graph across the y-axis; $f(x) = -a^x$ reflects across the x-axis.

Example: $y = 2^{x-2}$

• This is the graph of $y = 2^x$ shifted right 2 units.

Example: $y = 5 - 0.5^x$

• This graph is a reflection of $y = 2^x$ across the y-axis, then across the x-axis, and then shifted up 5 units.

Applications of Exponential Functions

Compound Interest Formula

Exponential functions are used to model compound interest, where the amount of money $A$ after $t$ years, with principal $P$, annual interest rate $r$ (in decimal), compounded $n$ times per year, is given by:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

Example: Investment Growth

• Suppose $100,000 is invested at 6.5% interest, compounded semiannually ($n = 2$).

• Function for amount after $t$ years: $A(t) = 100,000 \left(1 + \frac{0.065}{2}\right)^{2t} = 100,000(1.0325)^{2t}$

• Graphing: Use a graphing calculator with a suitable window, e.g., $[0, 30]$ for $t$ and $[0, 500,000]$ for $A$.

• Calculating values:

  • $A(0) = 100,000$

  • $A(4) \approx 129,157.75$

  • $A(8) \approx 166,817.25$

  • $A(10) \approx 189,583.79$

• Finding when $A = 400,000$: Solve $100,000(1.0325)^{2t} = 400,000$ for $t$ using a graphing calculator or algebraically. The solution is $t \approx 21.67$ years.

The Number e

Definition and Properties

• Euler's Number: $e$ is an irrational constant approximately equal to 2.7182818284...

• It is the base of the natural exponential function and arises in many mathematical contexts, especially in continuous growth and decay.

Examples of Calculating $e^x$

Expression	Value (4 decimal places)
$e^3$	20.0855
$e^{-0.23}$	0.7945
$e^0$	1
$e^1$	2.7183

Graphs of Exponential Functions with Base e

Comparing $f(x) = e^x$ and $g(x) = e^{-x}$

• $f(x) = e^x$ increases rapidly as $x$ increases.

• $g(x) = e^{-x}$ decreases rapidly as $x$ increases; it is a reflection of $f(x)$ across the y-axis.

x	$e^x$	$e^{-x}$
-3	0.04979	20.086
-2	0.13534	7.3891
-1	0.36788	2.7183
0	1	1
1	2.7183	0.36788
2	7.3891	0.13534
3	20.086	0.04979

Transformations with Base e

• Horizontal Translation: $f(x) = e^{x+3}$ is a translation of $y = e^x$ left 3 units.

• Horizontal Stretch and Reflection: $f(x) = e^{-0.5x}$ is a horizontal stretch of $y = e^x$ followed by a reflection across the y-axis.

• Horizontal Shrink, Reflection, and Translation: $f(x) = 1 - e^{-2x}$ is a horizontal shrink of $y = e^x$, reflected across the y-axis and x-axis, then translated up 1 unit.

Summary Table: Transformations of Exponential Functions

Transformation	Equation	Description
Horizontal Shift	$f(x) = a^{x-h}$	Right by $h$ units
Vertical Shift	$f(x) = a^x + k$	Up by $k$ units
Reflection (y-axis)	$f(x) = a^{-x}$	Reflect across y-axis
Reflection (x-axis)	$f(x) = -a^x$	Reflect across x-axis
Horizontal Stretch/Shrink	$f(x) = a^{bx}$	Stretch if $0 < b < 1$, shrink if $b > 1$

Additional info: Exponential functions are essential in modeling real-world phenomena such as population growth, radioactive decay, and financial investments. Mastery of their properties and transformations is foundational for further study in algebra and calculus.