Practice Maths

Topic Review — Exponential Functions

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★ U2T1 — Exponential Functions Review

Index laws • Graphs of exponential functions • Solving exponential equations • Growth and decay applications

  1. Q1 — Index laws

    Simplify: (a) x3 × x5   (b) (2a3)4   (c) 6x4 ÷ (2x7)   (d) (x2y3)0   (e) 4−3/2

  2. Q2 — Scientific notation

    (a) Write 0.000047 in scientific notation.   (b) Write 3.2 × 105 as an ordinary number.   (c) Calculate (4×103) × (2×104).

  3. Q3 — Graph features

    For y = 3 × 2x: (a) state the y-intercept and horizontal asymptote   (b) is it increasing or decreasing?   (c) what is y when x = 3? When x = −2?

  4. Q4 — Solving by matching bases

    Solve: (a) 2x = 128   (b) 9x+1 = 27   (c) 52x−1 = 1/25   (d) (1/4)x = 32

  5. Q5 — Growth model

    A town's population grows by 3% annually from a base of 25 000. (a) Write a model P(t). (b) Find the population after 8 years. (c) When does it exceed 40 000 (use trial and improvement)?

  6. Q6 — Decay and half-life

    A substance has half-life 10 years, initial mass 160 g. (a) Write a model. (b) Find mass after 30 years. (c) When does mass fall below 5 g?

  7. Q7 — Compound interest

    $4 000 invested at 5% p.a. compounded quarterly. (a) Write the model. (b) Find the balance after 6 years. (c) How many years to reach $6 000?

  8. Q8 — Comparing growth/decay

    Two populations: A(t) = 1000(1.05)t and B(t) = 2000(0.97)t. (a) Which is growing, which decaying? (b) Find when A = B (to nearest year using trial and improvement). (c) After 20 years, which is larger?

  9. Q9 — Depreciation

    A car bought for $35 000 depreciates at 18% per year. (a) Write a model V(t). (b) Find the value after 4 years. (c) After how many years is it worth less than $10 000?

  10. Q10 — Fitting and interpreting models

    Data: (0, 50), (2, 200). (a) Find a and b in A = a × bt. (b) Predict A at t = 5. (c) What does the rate of change of A at t = 0 suggest about early growth?

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