Understanding Exponential Functions and Their Graphs

Consider the function $y = 2^x$. We can observe its behavior through the following points:

• When $x = -1$, $y = 2^{-1} = \frac{1}{2}$
• When $x = 0$, $y = 2^0 = 1$
• When $x = 1$, $y = 2^1 = 2$
• When $x = 2$, $y = 2^2 = 4$

As $x$ becomes more negative, the curve approaches the x-axis but never actually touches or crosses it. This boundary is known as an asymptote.

Exam Tip: Always look for the y-intercept. For basic exponential functions in the form $y = b^x$, the graph will always pass through the point $(0, 1)$ because any non-zero number raised to the power of zero is one.

Exponential Decay ($0 < a < 1$)

When the base is a fraction between 0 and 1, the function represents exponential decay. In this case, as $x$ increases, the value of $y$ decreases, approaching zero.

The Decay Curve

The graph of a decay function is essentially a reflection of a growth function. While a growth curve rises steeply to the right, a decay curve falls steeply as it moves toward the right side of the coordinate plane.

For a function with a base like $0.1$, the values drop rapidly:

• As $x$ moves from negative to positive, the $y$ values shrink toward the x-axis.
• The x-axis serves as the horizontal asymptote for these curves as well.

Mathematical Properties and Calculations

Working with exponential functions often requires manipulating exponents, especially when dealing with negative exponents or fractions.

Negative Exponents

A negative exponent indicates a reciprocal. For example:

$$(0.1)^{-3} = \frac{1}{(0.1)^3} = \frac{1}{0.001} = 1000$$

Note that in handwritten study notes, errors can sometimes occur in manual calculations—such as miscalculating $2^{-2}$ or complex divisions involving $(0.1)^n$. It is vital to double-check these steps using the rule:

$$a^{-n} = \frac{1}{a^n}$$

Summary of Graph Characteristics

By mastering the relationship between the base, the table of values, and the resulting curve on the coordinate plane, you can accurately predict the behavior of any exponential model.