L07 — Exponential Functions
Key Terms
- Exponential function — a function of the form y = ax, where a > 0 and a ≠ 1
- Base a — the positive constant being raised to the power; controls growth or decay
- Exponential growth — when a > 1; the function increases rapidly as x increases
- Exponential decay — when 0 < a < 1; the function decreases towards zero as x increases
- Horizontal asymptote — the line y = 0; the curve approaches but never reaches the x-axis
- y-intercept — always (0, 1) for y = ax since a0 = 1 for any base a
Exponential Functions
| Form | Condition | Behaviour |
|---|---|---|
| y = ax | a > 1 | Exponential growth — increases as x increases |
| y = ax | 0 < a < 1 | Exponential decay — decreases as x increases |
Key Features of y = ax
| Feature | Value / Description |
|---|---|
| y-intercept | (0, 1) — since a0 = 1 for any base a |
| x-intercept | None — the curve never crosses the x-axis |
| Horizontal asymptote | y = 0 — the curve approaches but never reaches the x-axis |
| Domain | All real numbers (−∞, ∞) |
| Range | (0, ∞) — always positive |
Transformations
| Equation | Change from y = ax | New y-intercept | New asymptote |
|---|---|---|---|
| y = ax + c | Shift up c units | (0, 1+c) | y = c |
| y = b ⋅ ax | Vertical stretch by b | (0, b) | y = 0 |
| y = ax−h | Shift right h units | (0, a−h) | y = 0 |
| y = −ax | Reflection in x-axis | (0, −1) | y = 0 (from below) |
Real-World Connections
- Growth (a > 1): population growth, compound interest, bacteria doubling
- Decay (0 < a < 1): radioactive decay, drug clearance, cooling
- General form: A = A0 ⋅ at where A0 is the initial amount
What is an Exponential Function?
An exponential function has the form y = ax, where a is a fixed positive base (a ≠ 1) and x is the variable in the exponent. Unlike polynomial functions where x is the base, here x is the power. This makes exponential functions grow (or shrink) far more rapidly than any polynomial.
Key facts true for any base:
- a0 = 1, so the graph always crosses the y-axis at (0, 1).
- ax > 0 for all x, so the graph is always above the x-axis.
- The x-axis (y = 0) is a horizontal asymptote — the curve gets closer and closer but never touches it.
For y = 3x, complete the table and state the y-intercept, asymptote, and whether the function shows growth or decay.
| x | −2 | −1 | 0 | 1 | 2 |
| y | 1⁄9 | 1⁄3 | 1 | 3 | 9 |
Base a = 3 > 1 ⇒ exponential growth.
y-intercept: (0, 1).
Horizontal asymptote: y = 0.
Domain: all real numbers. Range: (0, ∞).
Growth vs Decay
The base a determines the behaviour:
- If a > 1: the output increases as x increases — exponential growth. The larger the base, the steeper the rise for positive x.
- If 0 < a < 1: the output decreases as x increases — exponential decay. Note that y = (0.5)x is the same as y = 2−x, so decay is just growth reflected in the y-axis.
Classify each function and state its y-intercept and asymptote.
(a) y = 5x (b) y = (0.4)x (c) y = 2x − 4
Solution:
(a) Base = 5 > 1 ⇒ growth. y-intercept: (0, 1). Asymptote: y = 0.
(b) Base = 0.4 < 1 ⇒ decay. y-intercept: (0, 1). Asymptote: y = 0.
(c) Base = 2 > 1 ⇒ growth. y-intercept: (0, 1−4) = (0, −3). Asymptote shifts to y = −4.
Transformations of y = ax
The same transformation rules that apply to other functions also apply to exponentials. The key is tracking how the y-intercept and the asymptote change.
For y = 3 × 2x + 1, state the y-intercept, horizontal asymptote, and describe the transformations applied to y = 2x.
Solution:
Start with y = 2x (y-intercept (0,1), asymptote y = 0).
×3 stretches vertically: y-intercept becomes (0, 3).
+1 shifts up: y-intercept becomes (0, 4); asymptote shifts to y = 1.
So: y-intercept = (0, 4), asymptote = y = 1.
The curve is steeper than y = 2x and sits above y = 1 for all x.
Real-World Applications
Exponential functions model any quantity that grows or decays by a constant percentage per unit of time — rather than by a constant amount. The general model is:
A = A0 × at
where A0 is the initial amount, a is the growth/decay factor, and t is time.
A colony starts with 200 bacteria and triples every hour.
(a) Write the equation for the population P after t hours.
(b) How many bacteria are there after 4 hours?
(c) After how many hours does the population first exceed 5000?
Solution:
(a) A0 = 200, a = 3 ⇒ P = 200 × 3t
(b) t = 4: P = 200 × 34 = 200 × 81 = 16 200 bacteria
(c) Try values: t=3: P=200×27=5400 > 5000. t=2: P=200×9=1800 < 5000.
⇒ After 3 hours the population first exceeds 5000.
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Key features from equations. Fluency
Complete the table. For each function, identify the base, behaviour (growth or decay), y-intercept, and horizontal asymptote.
Equation Base Growth or Decay? y-intercept Asymptote y = 3x y = (½)x y = 2 ⋅ 3x y = 5x + 2 -
y = 4x — evaluate and solve. Fluency
- (a) Complete the table of values for y = 4x:
x −2 −1 0 1 2 y - (b) Find x when y = 64. (Hint: express 64 as a power of 4.)
- (c) Find x when y = 1/64.
- (d) Does the graph of y = 4x ever cross the x-axis? Explain why or why not.
- (a) Complete the table of values for y = 4x:
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Growth or decay? Fluency
For each function, state whether it shows growth or decay, and find the y-intercept and horizontal asymptote.
- (a) y = 4x
- (b) y = (0.5)x
- (c) y = 3−x
- (d) y = 2x − 3
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Transformations — y-intercept and asymptote. Fluency
State the y-intercept and horizontal asymptote for each function.
- (a) y = 2x + 5
- (b) y = 3x − 1
- (c) y = 4 × 2x
- (d) y = 2x+1
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Find the equation y = ax. Understanding
Find the value of a and write the equation of the exponential function y = ax that passes through the given point.
- (a) (2, 9)
- (b) (3, 64)
- (c) (2, ¼)
- (d) (−1, 3)
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Bacteria growth. Understanding
A colony of bacteria starts with 500 bacteria and doubles every hour. The number of bacteria after t hours is P = 500 × 2t.
- (a) How many bacteria are there after 3 hours?
- (b) How many bacteria are there after 6 hours?
- (c) After how many hours will the population reach 8000? (Find t such that 500 × 2t = 8000.)
- (d) By what factor does the population increase in the first 4 hours?
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Drug clearance. Understanding
A patient takes 200 mg of a drug. The amount remaining in the bloodstream is halved every 4 hours, so A = 200 × (0.5)t/4, where t is hours.
- (a) How much remains after 4 hours?
- (b) How much remains after 12 hours?
- (c) After how many hours does the amount fall below 50 mg? (Use trial values of t = 4, 8, 12.)
- (d) As t → ∞, what happens to A? What is the horizontal asymptote, and what does it mean in context?
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Read and compare two exponential graphs. Understanding
The graph shows two exponential functions: one is y = 2x and the other is y = 3x.
- (a) Both graphs pass through the same point. Read its coordinates from the graph and explain why this makes sense algebraically.
- (b) For x > 0, which graph rises more steeply — y = 2x or y = 3x? Identify which curve is which from the graph.
- (c) For x < 0, which graph is higher? Read approximate values at x = −2 from the graph to support your answer.
- (d) State the horizontal asymptote common to both graphs and describe what it means for both functions.
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Compound interest. Problem Solving
$2000 is invested at 6% per annum compound interest. The value after t years is A = 2000 × 1.06t.
- (a) What is the value after 1 year?
- (b) What is the value after 10 years? (Give your answer to the nearest cent.)
- (c) By trial and error, find after how many complete years the investment first exceeds $4000. (Hint: work out when 1.06t > 2.)
- (d) If instead the interest rate were 8%, write the new formula. Would doubling take more or fewer years than at 6%?
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Coffee cooling. Problem Solving
A cup of coffee cools according to T = 20 + 70 × (0.9)t, where T is the temperature (°C) and t is minutes.
- (a) What is the initial temperature of the coffee (when t = 0)?
- (b) What is the temperature after 5 minutes? (Round to the nearest degree.)
- (c) As t → ∞, what temperature does the coffee approach? What does this represent in context?
- (d) By trial, estimate how many minutes it takes for the coffee to cool to below 48°C. (Try t = 8 and t = 9.)