📝 Chapter 3: Exponential Functions

Simplify: \( \dfrac{(3a^2)^3 \cdot a^{-4}}{a^3} \).

Evaluate \( 81^{3/4} \) without a calculator.

Evaluate \( 16^{-3/2} \) as a fraction.

For \( y = 4 \cdot 2^x \), state: (a) the y-intercept, (b) the horizontal asymptote, (c) whether it represents growth or decay.

Sketch \( y = (\tfrac{1}{2})^x \) for \( x \in \{-2, -1, 0, 1, 2\} \) using a table of values. State the range.

Determine whether each table represents linear, quadratic, or exponential. Justify with finite differences or the constant-ratio test:
(i) \( y \): 2, 5, 10, 17, 26    (ii) \( y \): 3, 6, 12, 24, 48    (iii) \( y \): 1, 4, 7, 10, 13

Given \( y = a \cdot b^x \) passes through \( (0, 6) \) and \( (3, 48) \). Find \( a \) and \( b \). Show all algebraic steps.

Sketch \( y = -2(3)^{x-1} + 4 \) by transforming the parent \( y = 3^x \). State each transformation, the asymptote, and the y-intercept.

Simplify completely: \( \dfrac{(2a^2 b^{-3})^{-2}}{(a^{-1} b^4)^3} \). Express your final answer with positive exponents only.

Explain why \( 2^0 = 1 \) using the quotient law of exponents. Then explain why \( 2^{-3} = \tfrac{1}{8} \).

Compare and contrast linear, quadratic, and exponential growth. Use a numerical example to show how exponential growth eventually dominates polynomial growth.

Describe the role of each parameter \( a, b, c \) in \( y = a \cdot b^{x} + c \). Use \( y = 3 \cdot 2^x - 1 \) as your worked example.

Why must the base of an exponential function satisfy \( b > 0, b \neq 1 \)? Explain what goes wrong if \( b = 1 \) or if \( b < 0 \).

A bacteria culture starts with 100 cells and triples every 4 hours. (a) Write the function \( N(t) \) for the population after \( t \) hours. (b) Find the population after 12 hours. (c) After 24 hours.

A radioactive element has half-life 5 days. A sample contains 80 grams. (a) Write \( A(t) \) for the mass remaining after \( t \) days. (b) Find the mass after 15 days. (c) Find the mass after 30 days.

A car worth \$24{,}000 depreciates 15% per year. (a) Write \( V(t) \) for the value after \( t \) years. (b) Find the value after 5 years. (c) Estimate (using a table or graph) when the car is worth half its original value.