Welcome back to Professor Baker's Math Class! Today, we are shifting gears from polynomial functions to a very powerful concept in mathematics: Exponential Functions. These functions are essential for modeling real-world phenomena like population growth, radioactive decay, and compound interest.

1. Reviewing the Laws of Exponents

Before diving into functions, it is crucial to have a solid handle on how exponents behave. As we discussed in class, here are the fundamental rules you need to memorize:

  • Product Rule: $x^m \cdot x^n = x^{m+n}$
  • Quotient Rule: \(\frac{x^m}{x^n} = x^{m-n}\)
  • Power of a Power: $(x^m)^n = x^{mn}$
  • Power of a Product: $(xy)^m = x^m y^m$
  • Negative Exponents: $x^{-m} = \frac{1}{x^m}$ or $\frac{1}{x^{-m}} = x^m$
  • Zero Exponent: $x^0 = 1$ (for $x \neq 0$)
  • Rational Exponents: $x^{m/n} = \sqrt[n]{x^m}$

2. Defining the Exponential Function

What differentiates an exponential function from the algebraic functions we've seen before? It's all about the location of the variable.

  • In a quadratic function like $y = x^2$, the base is variable ($x$) and the exponent is constant ($2$).
  • In an Exponential Function like $f(x) = 2^x$, the base is constant ($2$) and the exponent is the variable ($x$).

The general form is:

$$f(x) = a^x$$

3. The Rules of the Base (Growth vs. Decay)

For a function to be truly exponential, the base $a$ has specific constraints. As noted in our lecture:

  1. The base $a$ cannot be 1.
  2. The base $a$ cannot be negative.

Depending on the value of $a$, the behavior of the graph changes dramatically:

  • Exponential Growth: If $a > 1$, the function rises rapidly as $x$ increases (e.g., $f(x) = 2^x$).
  • Exponential Decay: If $0 < a < 1$, the function decreases towards zero as $x$ increases (e.g., $f(x) = 0.6^x$ or $f(x) = (\frac{1}{2})^x$).

4. Evaluating and Graphing

When graphing these functions, we often create a table of values. For example, comparing $f(x)=2^x$ and $f(x)=4^x$, we see that while both pass through the y-intercept $(0,1)$, the function with the larger base ($4^x$) grows much steeper, much faster.

We also practiced evaluating functions using calculators, including those with the natural base $e$:

  • $f(x) = 2^{-3.1} \approx 0.1166$
  • $f(x) = e^{-0.4} \approx 0.6703$

5. Why can't the base be negative?

In class, we looked at a table for $f(x) = (-2)^x$. The results oscillate between positive and negative values (e.g., $(-2)^2 = 4$ but $(-2)^3 = -8$). This creates a scattered plot of points rather than a continuous curve, which is why we define exponential bases as positive numbers only.

Homework Assignment

To ensure you've mastered these concepts, please complete the following problems from the textbook (Pages 185-187):

  • #2 - 6 (Evens)
  • #24 - 28 (Evens)
  • #32, 34, 71, 74, 76

Keep practicing those exponent rules—they are the key to unlocking this chapter! See you in the next class.