What is an exponential function?
An exponential function has the variable in the power: y = a · bˣ. Here a is the starting value and b is the base or growth factor. We need b > 0 and b ≠ 1.
In a table, when x goes up by 1, y is multiplied by the same number b. So the ratio of next y ÷ this y is constant. This is how you spot an exponential relation in data.
Exponent rules help simplify: bᵐ·bⁿ = bᵐ⁺ⁿ, (bᵐ)ⁿ = bᵐⁿ, b⁰ = 1, b⁻ⁿ = 1/bⁿ, b^(1/n) = ⁿ√b.
Graph, domain, range and asymptote
For y = bˣ:
- It passes through (0, 1), because b⁰ = 1.
- Domain: all real numbers. Range: y > 0.
- The x-axis (y = 0) is a horizontal asymptote: the curve gets close but never touches.
- b > 1: increasing (growth). 0 < b < 1: decreasing (decay).
- No x-intercept; y-intercept is 1 (or a for y = a·bˣ).
Linear, quadratic or exponential? Use differences
Look at a table with equal x-steps:
- First differences constant → linear.
- Second differences constant → quadratic.
- Ratios constant (and differences keep growing by the same ratio) → exponential.
Exponential growth is slow at first but, in the long run, beats any linear or quadratic function.
Transformations: y = a·b^(k(x − d)) + c
- a: vertical stretch (|a| > 1) or compression (|a| < 1); a < 0 reflects in the x-axis.
- k: horizontal compression by 1/|k|; k < 0 reflects in the y-axis.
- d: shifts right by d (left if d is negative).
- c: shifts up by c. The asymptote moves to y = c and the range becomes y > c (if a > 0).
Order to apply: stretches/reflections first, then shifts. For each point (x, y) of y = bˣ, the new point is (x/k + d, a·y + c).
Solving problems: growth, decay and data
- Growth by r %: A = P(1 + r)ᵗ. Compound interest n times a year: A = P(1 + r/n)ⁿᵗ.
- Doubling: N = N₀ · 2^(t/T), T = doubling time.
- Half-life: A = A₀ · (½)^(t/h), h = half-life.
- Decay by r %: A = P(1 − r)ᵗ (e.g. car value, cooling).
To model data: check that the ratios are roughly constant, find a from the start value and b from the average ratio, or use a graphing tool for exponential regression. Read answers off a graph by finding where the curve reaches a y-value.
Try it: fold a sheet of paper in half again and again. Count the layers: 1, 2, 4, 8, 16... After 7 folds you have 128 layers. That is 2ᵗ.
Key formulas and definitions
- y = a · bˣ (a ≠ 0, b > 0, b ≠ 1)
- y = a · b^(k(x − d)) + c (asymptote y = c)
- A = P(1 + r)ᵗ growth; A = P(1 − r)ᵗ decay
- A = P(1 + r/n)ⁿᵗ compound interest
- A = A₀ (½)^(t/h) half-life; N = N₀ · 2^(t/T) doubling
Worked examples
1. Is this table linear, quadratic or exponential? x: 0,1,2,3; y: 5, 15, 45, 135.
Differences: 10, 30, 90 (not constant). Ratios: 15/5 = 3, 45/15 = 3, 135/45 = 3 → constant ratio. Exponential: y = 5·3ˣ.
2. State the domain, range, asymptote and y-intercept of y = 2·3ˣ − 4.
Domain: all real x. a = 2 > 0, c = −4, so range y > −4. Asymptote y = −4. y-intercept: x = 0 → 2·1 − 4 = −2.
3. Describe the transformations of y = 2ˣ that give y = −3·2^(x − 1) + 5.
a = −3: vertical stretch by 3 and reflection in the x-axis. d = 1: shift right 1. c = 5: shift up 5. New asymptote y = 5, range y < 5.
4. ₹10,000 (or any currency) is invested at 8 % per year compounded yearly. Find the amount after 5 years.
A = P(1 + r)ᵗ = 10000 × 1.08⁵. 1.08² = 1.1664; 1.08⁴ = 1.3605; ×1.08 = 1.4693. A ≈ 14,693.
5. A radioactive sample of 80 g has a half-life of 6 days. How much is left after 18 days?
Number of half-lives = 18 ÷ 6 = 3. A = 80 × (½)³ = 80 ÷ 8 = 10 g.
6. A bacteria culture starts with 200 cells and doubles every 3 hours. When will it reach 6400 cells?
6400 = 200 · 2^(t/3) → 32 = 2^(t/3). 32 = 2⁵, so t/3 = 5 → t = 15 hours.
Common mistakes
- Writing 2·3ˣ as 6ˣ. The power applies only to the base 3: 2·3² = 18, but 6² = 36.
- Thinking the graph crosses the x-axis. y = a·bˣ with a > 0 is always positive; the x-axis is an asymptote.
- Using r = 8 instead of 0.08 in (1 + r)ᵗ. Change the percentage to a decimal first.
- Mixing up d's sign: y = 2^(x − 3) moves RIGHT 3, not left.