Identifying Exponential Functions

Determine which function is exponential: $f(x) = 3x^2$, $g(x) = 2^x$, $h(x) = x^3$

Check $f(x) = 3x^2$: The variable $x$ is in the base, raised to power 2 = Quadratic (polynomial), not exponential

Check $g(x) = 2^x$: The variable $x$ is in the exponent, base is constant 2 = This IS exponential: $a = 1$, $b = 2$

Check $h(x) = x^3$: The variable $x$ is in the base, raised to power 3 = Cubic (polynomial), not exponential

Answer: $g(x) = 2^x$ is the exponential function because the variable is in the exponent.

Graphing an Exponential Growth Function

Graph $f(x) = 2^x$ by creating a table of values.

Calculate $f(-2)$: $f(-2) = 2^{-2} = \frac{1}{2^2} = \frac{1}{4} = 0.25$ = Point: $(-2, 0.25)$

Calculate $f(-1)$: $f(-1) = 2^{-1} = \frac{1}{2} = 0.5$ = Point: $(-1, 0.5)$

Calculate $f(0)$: $f(0) = 2^0 = 1$ = Point: $(0, 1)$ — the y-intercept

Calculate $f(1)$: $f(1) = 2^1 = 2$ = Point: $(1, 2)$

Calculate $f(2)$: $f(2) = 2^2 = 4$ = Point: $(2, 4)$

Calculate $f(3)$: $f(3) = 2^3 = 8$ = Point: $(3, 8)$

Describe the graph: As $x$ increases, $y$ doubles. As $x$ decreases, $y$ approaches 0 but never reaches it. = Exponential growth curve

Answer: The graph passes through $(0,1)$, rises steeply to the right, and approaches the x-axis (asymptote) to the left.

Exponential Decay Function

A radioactive substance decays according to $A(t) = 100 \cdot (0.5)^t$, where $t$ is time in hours. Find the amount remaining after 3 hours.

Identify the components: Initial amount $a = 100$, decay factor $b = 0.5$, time $t = 3$ = This is exponential decay since $0 < b < 1$

Substitute $t = 3$: $A(3) = 100 \cdot (0.5)^3$ = $A(3) = 100 \cdot (0.5)^3$

Calculate $(0.5)^3$: $(0.5)^3 = 0.5 \times 0.5 \times 0.5 = 0.125$ = $(0.5)^3 = 0.125$

Multiply by 100: $A(3) = 100 \cdot 0.125 = 12.5$ = 12.5 grams remain

Answer: After 3 hours, 12.5 grams of the substance remain. Each hour, half of the substance decays.

Compound Interest

You invest 1000 dollars at 5% annual interest, compounded annually. Write the function and find the value after 10 years.

Write the compound interest formula: $A(t) = P(1 + r)^t$ where $P$ = principal, $r$ = rate, $t$ = years = Exponential growth model

Identify values: $P = 1000$, $r = 0.05$, so $b = 1 + 0.05 = 1.05$ = $A(t) = 1000 \cdot (1.05)^t$

Calculate for $t = 10$: $A(10) = 1000 \cdot (1.05)^{10}$ = Need to compute $(1.05)^{10}$

Evaluate $(1.05)^{10}$: $(1.05)^{10} \approx 1.6289$ = Growth multiplier after 10 years

Final calculation: $A(10) = 1000 \cdot 1.6289 \approx 1628.89$ = Approximately 1628.89 dollars

Answer: After 10 years, the investment grows to approximately 1628.89 dollars — a gain of about 629 dollars from interest alone.

Mistake: Confusing $2^x$ with $x^2$

Why: In $2^x$, the variable is the exponent (exponential). In $x^2$, the variable is the base (polynomial).

Correct: Look at WHERE the variable is: exponent = exponential function, base = polynomial function.

Mistake: Thinking the graph can cross or touch the x-axis

Why: Since $b^x > 0$ for all real $x$ (when $b > 0$), and $a \cdot b^x$ preserves this (for $a > 0$), the function never equals zero.

Correct: The x-axis is a horizontal asymptote. The graph approaches it infinitely but never reaches it.

Mistake: Forgetting that $b^0 = 1$, not 0

Why: Any non-zero number raised to the power 0 equals 1. This is why exponential functions pass through $(0, a)$.

Correct: $f(0) = a \cdot b^0 = a \cdot 1 = a$. The y-intercept is always $a$.

Mistake: Using a negative base

Why: Negative bases cause problems: $(-2)^{0.5}$ is not a real number. We restrict to $b > 0$.

Correct: The base $b$ must be positive. For decay, use $0 < b < 1$, not negative numbers.

Compound Interest in Banking

Banks use exponential functions to calculate how your savings grow over time with compound interest.

With 5% annual compound interest, 1000 dollars becomes $1000 \cdot (1.05)^{20} \approx 2653$ dollars after 20 years.

Population Growth

Biologists model population growth using exponential functions when resources are unlimited.

A bacteria colony that doubles every hour follows $P(t) = P_0 \cdot 2^t$. Starting with 100 bacteria, after 8 hours there are $100 \cdot 2^8 = 25,600$ bacteria.

Radioactive Decay in Medicine

Medical imaging uses radioactive tracers that decay exponentially, allowing doctors to track their movement through the body.

Technetium-99m has a half-life of 6 hours. If you start with 100 mg, after 18 hours (3 half-lives) you have $100 \cdot (0.5)^3 = 12.5$ mg remaining.

Q: What is the difference between exponential and polynomial functions?

A: In exponential functions like $2^x$, the variable is in the exponent. In polynomial functions like $x^2$, the variable is in the base. This makes exponential functions grow much faster for large $x$.

Q: Why must the base be positive?

A: Negative bases cause problems with non-integer exponents. For example, $(-4)^{0.5} = \sqrt{-4}$, which is not a real number. We restrict to $b > 0$ to keep outputs real.

Q: What does the number $e \approx 2.718$ have to do with exponential functions?

A: The constant $e$ is the base of the natural exponential function $f(x) = e^x$. It appears naturally in calculus and is used in continuous growth models like continuously compounded interest.

Q: Can exponential decay ever reach zero?

A: Mathematically, no. The function approaches zero but never reaches it (asymptotic behavior). In practice, we might consider the quantity negligible after many half-lives.