Exponential Functions | PreCalculus

About this Video

• Video Title: Exponential Functions | PreCalculus
• Channel: Professor Mark Gutierrez
• Speakers: Professor Mark Gutierrez
• Duration: 01:28:08

Introduction

This PreCalculus video from Professor Mark Gutierrez explains exponential functions, their properties, and applications in finance (simple and compound interest calculations, including continuous compounding). The video focuses on understanding the definition of exponential functions, the restrictions on the base, graphing exponential functions, identifying key features of their graphs (domain, range, asymptotes, intercepts), and applying exponential functions to financial calculations.

Key Takeaways

1. Definition of Exponential Function: An exponential function is of the form f(x) = bˣ, where the base 'b' is a real number, strictly greater than 0 and not equal to 1, and 'x' is any real number.
2. Restrictions on the Base: The base (b) cannot be negative (to avoid imaginary solutions) and cannot be 1 (to ensure the function is one-to-one and has an inverse).
3. Graphing Exponential Functions: Exponential functions have a characteristic curve that approaches a horizontal asymptote. The y-intercept is always (0, 1), and the point (1, b) is always on the curve. Transformations (shifts, reflections, stretches) affect the graph's position but not its basic shape.
4. Horizontal Asymptotes: The horizontal asymptote of a basic exponential function is y = 0. Vertical translations shift the asymptote up or down; the asymptote's equation is always y = c, where 'c' is the constant term in the function.
5. Financial Applications: The video demonstrates applying exponential functions to calculate simple interest (I = prt) and compound interest (A = P(1 + r/n)^(nt)), including continuous compounding (A = Pe^(rt)). Compounding more frequently results in larger account balances.