zuai-logo

Exponential graphs

Lisa Chen

Lisa Chen

6 min read

Listen to this study note

Study Guide Overview

This guide covers exponential graphs, including their basic structure (initial value, base, domain, range), growth and decay patterns, and real-world applications. It explains how to interpret these graphs, determine their equations (finding initial value and base), and calculate doubling time/half-life. The guide also provides practice questions and emphasizes key exam topics like growth/decay, graph interpretation, and equation determination.

Exponential Graphs: Your Ultimate Guide 🚀

Hey there, future AP superstar! Exponential graphs might seem intimidating, but they're actually super cool and totally manageable. Think of them as the superheroes of functions, growing or shrinking at crazy speeds. This guide will break it all down so you’re feeling confident and ready to ace any question on test day! Let's dive in!

Memory Aid

Remember the basic form: f(x) = a * b^x. Think of 'a' as your starting point and 'b' as the growth/decay factor.

Characteristics of Exponential Graphs

Basic Structure and Components

  • Exponential functions follow the form: f(x)=a∗bxf(x) = a * b^x

  • a: The initial value (y-intercept). It's where the graph starts on the y-axis.

  • b: The base (growth or decay factor). This determines how quickly the graph changes.

  • Domain: All real numbers (x can be anything!).

  • Range: Positive real numbers (assuming a > 0). The graph will never touch or cross the x-axis.

  • Y-intercept: Always equals the initial value a. When x=0, f(0)=a∗b0=af(0) = a * b^0 = a.

  • The rate of change is proportional to the current function value, which gives the graph its characteristic curve.

Key Concept

The y-intercept is your starting point, and the base dictates the curve's direction (up for growth, down for decay).

Growth and Decay Patterns

  • Exponential Growth: Occurs when b > 1. The graph shoots upwards as x increases. 📈
  • Exponential Decay: Occurs when 0 < b < 1. The graph decreases, approachin...