Precalculus is largely about the structure of the Elementary Functions and how we reason about them. This includes power functions; radical and root functions; linear, quadratic, and other polynomial functions; rational functions; exponential functions; logarithmic functions; and a library of trigonometric functions (which are introduced in this course).

Our focus is analyzing these functions.

• Domain
• Zeros

• Continuity

  • discontinuities
  • singularities
• End-Behavior

• Behavior

  • intervals where increasing
  • intervals where decreasing
• Global Maximum and Minimum
• Local Maximums and Minimums
• Range
• ...and we would like a nice graph

As a segue from the first half of Precalculus, let’s quickly review our growing library of Elementary Functions. We’ll begin with the exponential functions.

\(\blacktriangleright \) Basic Exponential Functions

The template for the basic exponential function looks like \(exp(x) = a \cdot r^x\).

• \(a\) is the leading coefficient. It primarily determines the sign of the function values.
• \(r\) is the base. It determines how the function values grow, depending on whether \(0 < r < 1\) or \(1 < r\).

\(\blacktriangleright \) Shifted Exponential Functions

The template for a shifted exponential function looks like \(exp(x) = a \cdot r^{b \, x + c} + d\) or just \(a \cdot r^x + d\).

• \(c\) describes a shift in the domain from the basic exponential function.
• \(d\) describes a shift in the function value from the basic exponential function.

The first term in the formula for a shifted exponential function is, by itself, an exponential function:

It CAN be written in the form \(A \cdot R^x\), where \(A = a \cdot r^c\) and \(R = r^b\).

Algebraically, the second term, the “+ d” cannot be absorbed into the exponential form. However, shifting functions is a very common operation and so, we’ll include \(a \cdot r^{b \, x + c} + d\) in our general ideas of exponential functions.

Learning Outcomes

In this section, students will

• describe the full picture of all exponential functions and their graphs.