Rucete ✏ AP Precalculus In a Nutshell
5. Exponential Functions
This chapter explores exponential functions, their graphs, transformations, real-world applications, and how to compare models. These functions model rapid growth or decay and are vital in many contexts including finance, population dynamics, and radioactive decay.
- Definition and Graph of Exponential Functions
- f(x) = abˣ, where a ≠ 0, b > 0, b ≠ 1
- Domain: all real numbers; Range: (0, ∞)
- y-intercept at (0, a); Horizontal asymptote: y = 0
- If b > 1 → exponential growth
- If 0 < b < 1 → exponential decay
- Always concave up; no extrema unless on closed interval
- End Behavior
- As x → ∞, f(x) → ∞ (for growth)
- As x → −∞, f(x) → 0
- The graph never touches the x-axis (asymptotic to y = 0)
- Laws of Exponents (Apply to Exponential Functions)
- ax · ay = ax+y
- (ax)y = axy
- ax / ay = ax−y
- a⁰ = 1
- a⁻ⁿ = 1/aⁿ
- a^1/n = ⁿ√a
- Graph Transformations
- f(x + h) → shift left h units
- f(x) + k → shift up k units
- −f(x) → reflect over x-axis
- f(−x) → reflect over y-axis
- a · f(x) → vertical stretch or compression
- Horizontal shift can often be written as a vertical dilation using exponent rules
- Modeling Real-World Data
- Exponential functions model repeated multiplication:
f(x) = abˣ, where a = initial value, b = growth/decay factor - Growth: b > 1
- Decay: 0 < b < 1
- Example (growth):
$450 compounded at 8.2% annually → f(t) = 450(1.082)ᵗ - Example (decay):
A car losing 11.75% per year → f(t) = 32750(0.8825)ᵗ
- Finding Equations from Data
- Use two data points:
Solve system to find a and b in f(x) = abˣ - Use graphing calculator regression:
- ExpReg function on TI graphing calculators
- Get best-fit curve and r²-value (closer to 1 = better fit)
- Continuous Growth and Decay
- Modeled by f(t) = aeᵏᵗ, where e ≈ 2.718
- k > 0 → growth; k < 0 → decay
- Example: $12,000 invested at 5% interest compounded continuously
→ f(t) = 12000e^0.05t
- Competing Model Validation
- Use regression (linear, quadratic, exponential) and choose the best fit based on:
- Scatterplot shape
- r²-value (coefficient of determination)
- Residual plots (random = good fit; curved = poor fit)
- Residual = actual value − predicted value
- Use calculator to plot residuals for visual model validation
- Real-World Examples
- Bacterial growth: P(t) = initial × growth⁽ᵗ⁾
- Half-life: N(t) = initial × (½)^(t/half-life)
- Car depreciation, population, interest, etc.
In a Nutshell
Exponential functions describe processes that change rapidly over time—whether growing or decaying. With a solid grasp of graph shapes, transformations, modeling techniques, and regression tools, you can confidently analyze real-world scenarios and determine the most accurate function model.