Exponential Functions

Understanding Exponential Functions

In Algebra II, exponential functions form one of the main families of functions you will study, alongside linear, polynomial, rational, and later logarithmic functions. Here you will focus on what makes exponential functions special, how they are written, and how their graphs behave. Detailed applications, growth, and decay are treated in the later subsections.

Basic Form and Definition

An exponential function is a function in which the variable appears in the exponent. The standard basic form is

$$
f(x) = a \cdot b^x
$$

where:

• $a$ is a nonzero constant called the initial value or starting amount.
• $b$ is a positive constant called the base of the exponential.
• $x$ is the variable (typically representing time or some input).
• We usually require $b>0$ and $b \neq 1$ to get a genuine exponential behavior.

Key point: In an exponential function, $x$ is in the exponent, not multiplied by it or added outside it.

Compare:

• Exponential: $f(x) = 2 \cdot 3^x$
• Not exponential: $g(x) = x^2$ (this is a power function: the variable is the base, not the exponent).

Growth vs. Decay

The base $b$ determines whether the function represents exponential growth or exponential decay.

• Exponential growth occurs when $b>1$.
• Example: $f(x) = 5 \cdot 2^x$ grows as $x$ increases.
• Exponential decay occurs when $0<b<1$.
• Example: $g(x) = 100 \cdot (0.8)^x$ decreases as $x$ increases.

The sign of $a$ affects whether the graph is above or below the $x$-axis, but not whether it is growth or decay in magnitude.

Key Features of the Graph

Consider $f(x) = a \cdot b^x$ with $a>0$ and $b>0$, $b\neq 1$.

Domain and Range

• Domain: All real numbers, $(-\infty, \infty)$.
• Range:
• If $a>0$, then $f(x)>0$ for all $x$, so the range is $(0,\infty)$.
• If $a<0$, then $f(x)<0$ for all $x$, so the range is $(-\infty,0)$.

Intercepts

• $y$-intercept: At $x=0$,
  $$
  f(0) = a \cdot b^0 = a \cdot 1 = a.
  $$
  So the graph always passes through $(0,a)$.
• $x$-intercept: For $a \neq 0$, $f(x) = a \cdot b^x$ is never $0$, so there is no $x$-intercept.

Asymptote

The graph of $f(x) = a \cdot b^x$ has a horizontal asymptote:

• For $a>0$, as $x \to -\infty$ (for growth functions) or $x \to \infty$ (for decay functions), the function values get closer and closer to $0$ but never reach it. So the $x$-axis, $y=0$, is a horizontal asymptote.
• If the function is shifted vertically by $k$ units, for example
  $$
  f(x) = a \cdot b^x + k,
  $$
  then the horizontal asymptote becomes $y = k$.

End Behavior

For $a>0$:

• If $b>1$ (growth):
• As $x \to \infty$, $f(x) \to \infty$.
• As $x \to -\infty$, $f(x) \to 0$.
• If $0<b<1$ (decay):
• As $x \to \infty$, $f(x) \to 0$.
• As $x \to -\infty$, $f(x) \to \infty$.

For $a<0$, these behaviors are reflected across the $x$-axis.

Graphing Basic Exponential Functions

To sketch $f(x) = a \cdot b^x$:

1. Plot the $y$-intercept at $(0,a)$.
2. Use another simple point, often $x=1$:
   $$
   f(1) = a \cdot b.
   $$
3. Consider the horizontal asymptote: for $a>0$ and no vertical shift, draw a dashed line along $y=0$ to guide the curve.
4. Shape:
• For $b>1$ and $a>0$, the curve increases from left to right, approaching $0$ on the left and rising steeply on the right.
• For $0<b<1$ and $a>0$, the curve decreases from left to right, starting large on the left and approaching $0$ on the right.

Only a few points and the asymptote are needed to draw the general exponential shape.

Transformations of Exponential Functions

You can transform the basic exponential graph in ways similar to other functions. Starting with a base function $f(x) = b^x$:

• Vertical stretch or compression: $y = a \cdot b^x$
• If $|a|>1$, the graph is stretched vertically.
• If $0<|a|<1$, the graph is compressed vertically.
• If $a<0$, it is reflected across the $x$-axis.
• Vertical shift: $y = b^x + k$
• Moves the graph up by $k$ if $k>0$, down by $k$ if $k<0$.
• Changes the horizontal asymptote from $y=0$ to $y=k$.
• Horizontal shift: $y = b^{x-h}$
• Moves the graph right by $h$ if $h>0$, left by $h$ if $h<0$.
• The $y$-intercept changes, but the asymptote stays at $y=0$ (unless there is also a vertical shift).
• Reflections horizontally: $y = b^{-x}$
• Reflects the graph of $y = b^x$ across the $y$-axis.
• Note that $b^{-x} = \left(\frac{1}{b}\right)^x$, so $y = b^{-x}$ is a decay function when $b>1$.

A general exponential function with transformations can be written as:

$$
y = a \cdot b^{x - h} + k
$$

where $a$, $b$, $h$, and $k$ affect the graph as described.

Recognizing Exponential Functions from Tables

From a table of values, you can recognize exponential behavior by looking for a constant multiplicative factor between $y$-values as $x$ increases by 1.

For example, if:

• When $x$ increases by 1, $y$ is always multiplied by $3$ (e.g., $2, 6, 18, 54, \dots$), then the function looks like $y = a \cdot 3^x$ for some $a$.
• This contrasts with linear functions, where the difference between $y$-values is constant instead of the ratio.

If the ratio
$$
\frac{y_{x+1}}{y_x}
$$
is constant (for evenly spaced $x$ values), that suggests an exponential function.

Exponential Functions and the Number $e$

Besides bases like $2$, $3$, and $10$, there is a special irrational number $e \approx 2.71828\ldots$ which is often used as a base:

$$
f(x) = a \cdot e^x.
$$

Functions with base $e$ play an important role in calculus and in natural growth and decay models. In this chapter, it is enough to recognize $e^x$ as an exponential function with all the same basic properties (domain, range, transformations) as $b^x$ for other positive bases $b\neq 1$.

Solving Simple Equations Involving Exponentials

Without using logarithms (covered in another chapter), you can solve some exponential equations by rewriting both sides with the same base.

For example, solve $2^x = 8$.

• Recognize $8 = 2^3$, so
  $$
  2^x = 2^3 \implies x = 3.
  $$

Similarly, for $5 \cdot 3^x = 45$:

• Divide both sides by $5$:
  $$
  3^x = 9.
  $$
• Recognize $9 = 3^2$:
  $$
  3^x = 3^2 \implies x = 2.
  $$

More complicated exponential equations that cannot be rewritten with a common base will be solved using logarithms in a separate chapter.

Comparing Exponential and Other Functions

Within Algebra II, it is useful to compare exponential functions with linear and polynomial functions:

• Linear functions grow by equal additive amounts over equal $x$-intervals.
• Exponential functions grow or decay by equal multiplicative factors over equal $x$-intervals.

As $x$ becomes large (for $b>1$), exponential functions eventually dominate the growth of any fixed-degree polynomial. This difference in long-term behavior is an important feature of exponentials.

Typical Algebra II Skills with Exponential Functions

In this chapter, the main skills you should aim to develop are:

• Identifying whether a function is exponential.
• Writing functions in the form $f(x) = a \cdot b^x$.
• Determining growth vs. decay from the base.
• Graphing exponential functions, including shifts and reflections.
• Finding domain, range, intercepts, and asymptotes of exponential functions.
• Recognizing exponential patterns in tables of values.
• Solving simple exponential equations by rewriting with a common base.

More specialized topics—such as detailed growth and decay formulas, percentage growth rates, continuous growth with base $e$, and solving exponential equations with logarithms—are treated in the subsequent sections on growth, decay, and logarithms.