AP Precalculus: Exponential Functions Formulas & Concepts

1. Exponential Function Form

General form: \( f(x) = ab^{x} \)
- \(a\) is the initial value or y-intercept.
- \(b\) is the base: \( b > 0, b \neq 1 \)

2. Domain and Range

• Domain: all real numbers (\( -\infty < x < \infty \))
• Range: positive values (\( (0, \infty) \)) if \(a > 0\); reversed if \(a < 0\)
• Asymptote: \( y = 0 \) (the x-axis)

3. Matching Exponential Functions and Graphs

• If \( b > 1 \): exponential growth (rises right, falls left)
• If \( 0 < b < 1 \): exponential decay (falls right, rises left)
• If \( a < 0 \): reflects graph about x-axis
• All have a horizontal asymptote at \( y = 0 \)

4. Linear vs. Exponential Functions

• Linear: \( f(x) = mx + b \)
  • Equal additive change per unit interval (constant difference)
• Exponential: \( f(x) = ab^x \)
  • Equal multiplicative change per unit interval (constant ratio)

5. Exponential Growth and Decay

• Growth: \( f(x) = ab^{x} \), \( b > 1 \) (percentage increase: \( b = 1 + r \), \( r>0 \))
• Decay: \( f(x) = ab^{x} \), \( 0 < b < 1 \) (percentage decrease: \( b = 1 - r \), \( r>0 \))
• Continuous Exponential Model: \( f(x) = ae^{kx} \); growth if \( k>0 \), decay if \( k<0 \)