Pre-Calculus - Week 4

Exponential Functions

Exponential functions model the most important real-world processes: population growth, radioactive decay, compound interest, and viral spread. Unlike polynomial growth, exponential growth accelerates continuously — doubling, then doubling again. This week you master graphing, transformations, and the number e, all of which reappear immediately in Calculus I when you differentiate exponential functions.

Learning Objectives

- Evaluate and simplify expressions with integer and rational exponents. - Define exponential functions f(x) = a^x and state the natural domain and range. - Identify and apply growth vs. decay based on the base a. - Graph exponential functions and describe their key features. - Apply transformations to exponential graphs. - Use the natural base e and understand its significance. - Solve exponential equations by matching bases or using logarithms. - Apply compound interest and continuous growth formulas.

1. Exponent Review

Key rules (a > 0, b > 0, m and n rational): a^m * a^n = a^(m+n) a^m / a^n = a^(m-n) (a^m)^n = a^(mn) (ab)^m = a^m * b^m a^0 = 1 a^(-n) = 1/a^n a^(1/n) = nth root of a a^(m/n) = (nth root of a)^m Example: Simplify (2^3 * 2^(-1))^2 = (2^2)^2 = 2^4 = 16. Example: 27^(2/3) = (cube root of 27)^2 = 3^2 = 9.

2. Defining Exponential Functions

An exponential function has the form f(x) = a^x where: - a > 0 (base must be positive) - a is not 1 (if a = 1, f(x) = 1 for all x — not very useful) - x is any real number Domain: all real numbers (-inf, inf). Range: (0, inf). Exponential functions are always positive. Two behaviors: - a > 1: GROWTH. f is increasing. As x goes to -inf, f approaches 0. As x goes to +inf, f goes to infinity. - 0 < a < 1: DECAY. f is decreasing. As x goes to +inf, f approaches 0.

3. The Natural Base e

The number e ≈ 2.71828... is defined as: e = lim n->inf (1 + 1/n)^n It arises naturally in continuous growth models. The function f(x) = e^x is THE exponential function of calculus because its derivative is itself: d/dx(e^x) = e^x. Practical values: e^0 = 1 e^1 ≈ 2.718 e^(-1) ≈ 0.368 e^2 ≈ 7.389 The function f(x) = e^x is increasing, has y-intercept (0, 1), and the x-axis (y = 0) is a horizontal asymptote as x goes to -infinity.

4. Graphing Exponential Functions

Key points for f(x) = a^x: f(-1) = 1/a, f(0) = 1, f(1) = a. The y-intercept is always (0, 1). Horizontal asymptote: y = 0. Example 1: f(x) = 2^x. Points: (-1, 1/2), (0, 1), (1, 2), (2, 4). Growing steeply to the right. Example 2: f(x) = (1/2)^x = 2^(-x). Same shape as 2^x, reflected over y-axis. Points: (-1, 2), (0, 1), (1, 1/2). Decaying.

5. Transformations of Exponential Graphs

Starting from f(x) = a^x: - f(x) + k: shift up k. - f(x) - k: shift down k. - f(x - h): shift right h. - f(x + h): shift left h. - -f(x): reflect over x-axis. - f(-x): reflect over y-axis (= decay if a was growth). Example 1: g(x) = 2^x + 3. Shifts graph up 3. HA moves to y = 3. Example 2: g(x) = e^(x-1) - 2. Shift right 1, down 2. HA: y = -2. y-intercept: e^(-1) - 2 ≈ -1.63.

6. Compound Interest Formula

If P dollars is invested at annual rate r (as a decimal), compounded n times per year for t years: A = P(1 + r/n)^(nt) Example: Invest $1000 at 6% annual rate, compounded monthly for 5 years. A = 1000(1 + 0.06/12)^(12*5) = 1000(1.005)^60 ≈ 1000 * 1.3489 ≈ $1348.85 Compared to simple interest: A = 1000(1 + 0.06*5) = $1300.

7. Continuous Compound Interest and e

When compounding is continuous (n goes to infinity): A = Pe^(rt) This is where e comes from in finance. Example: $1000 at 6% for 5 years continuously compounded: A = 1000 * e^(0.06*5) = 1000 * e^0.3 ≈ 1000 * 1.3499 ≈ $1349.86. The continuous model A = Pe^(rt) and the decay model A = Pe^(-rt) appear constantly in calculus (population models, Newton's law of cooling, radioactive decay).

8. Solving Exponential Equations by Matching Bases

If the equation can be written with the same base on both sides, set the exponents equal. Example 1: Solve 2^(x+1) = 8. 8 = 2^3. So 2^(x+1) = 2^3. Exponents must be equal: x + 1 = 3. x = 2. Example 2: Solve 9^x = 27. 9 = 3^2, 27 = 3^3. (3^2)^x = 3^3. 3^(2x) = 3^3. 2x = 3. x = 3/2. Example 3: Solve 4^(2x-1) = 64. 64 = 4^3. So 2x - 1 = 3. 2x = 4. x = 2.

9. Solving Exponential Equations with Logarithms

When bases cannot be matched, take the log of both sides. (Logarithms are covered in depth next week.) Example: Solve 5^x = 20. ln(5^x) = ln(20) x * ln(5) = ln(20) x = ln(20)/ln(5) ≈ 2.996/1.609 ≈ 1.861. Alternatively: log(5^x) = log(20). x*log(5) = log(20). x = log(20)/log(5). The two methods give the same answer (change-of-base formula).

10. Half-Life and Radioactive Decay

Radioactive decay follows: A(t) = A_0 * e^(-kt) or A(t) = A_0 * (1/2)^(t/h) where h is the half-life. Example: Carbon-14 has half-life ≈ 5730 years. A sample starts with 200 grams. A(t) = 200 * (1/2)^(t/5730). After 11460 years: A = 200 * (1/2)^2 = 200/4 = 50 grams. Decay constant: if half-life is h, then k = ln(2)/h. A(t) = A_0 * e^(-ln(2)*t/h).

11. Common Mistakes

- Thinking a^(-x) is negative: a^(-x) = 1/a^x, still positive. - Confusing growth and decay: a > 1 is growth, 0 < a < 1 is decay. - Writing e^2 as 2e (wrong): e^2 means e*e ≈ 7.389, not 2e. - Forgetting to transform the horizontal asymptote when shifting vertically. - In compound interest, using r as a percentage instead of a decimal.