Exponential Functions Exploration

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📚 Part 1: Multiple Choice & Quick Checks

1. For the function y = 4(1.2)^x, identify a and b.

a = 4, b = 1.2

a = 1.2, b = 4

a = 4, b = 0.2

2. Which of the following statements is true for a standard exponential function y = ab^x with a > 0 and b > 1?

The horizontal asymptote is y = 0 and the range is y > 0.

The function has a vertical asymptote at x = 0.

The domain is y > 0 and the range is all real numbers.

3. If f(x) = 5 × 2^x, what is f(0)?

4. Check all that apply: Which functions represent exponential decay?

g(x) = 7(0.9)^x

h(x) = 3(1.05)^x

k(x) = 2(1/2)^x

5. Evaluate f(x) = 5 × 2^x for these values: f(-1), f(1), f(3).

6. Solve each equation. Show working.

a) 3^x = 81

b) 2^x+1 = 16

7. Real-world context: A bacterial culture starts with 500 bacteria and doubles every 3 hours.

a) Write a function P(t) for the population after t hours.

b) Calculate the population after 6 hours.

c) Sketch P(t) for 0 ≤ t ≤ 9 hours. Label the axes and show key points.

8. Finance application: $1000 is invested at 5% p.a. compound interest. Use A = P(1 + r)^t.

a) Find the amount after 3 years.

b) Briefly explain how the base of the exponential reflects growth in this context (one or two sentences).