Mathematics Advanced • Year 11 • Module 4 • Lesson 2
Graphs of Exponential Functions
Apply transformations and equation-from-features techniques to real and abstract contexts.
Problem 1 — Cooling coffee (vertical translation)
A cup of coffee cools in a room at 20°C. Its temperature is modelled by
T(t) = 70 · (0.9)t + 20, t in minutes, T in °C
Set up: What are we solving for?
(i) State the asymptote of the T-graph and interpret it physically. 2 marks
(ii) Find T(0) and T(5) (1 d.p.). What temperature did the coffee start at? 2 marks
(iii) Sketch the temperature graph for 0 ≤ t ≤ 30, showing the asymptote, the y-intercept, and the value at t = 5. State the long-run temperature. 3 marks
Stuck? Revisit lesson § Worked Example 2 (translation example).Problem 2 — Fish population (find equation from data)
A new fish species is introduced to a lake. The population (in thousands) measured each year is modelled by y = a · bx. The data shows:
| Year x | Population y (thousands) |
|---|---|
| 0 | 5 |
| 2 | 20 |
Set up: What are we solving for?
(i) Use the (0, 5) data point to find a. 1 mark
(ii) Use the (2, 20) data point to find b (show one line of working). 2 marks
(iii) Use the model to predict the population at year 5. State your assumption (one sentence) about validity of the model for years beyond the data. 2 marks
Problem 3 — Drug clearance (decay graph)
A 200 mg dose of a drug is absorbed instantly. The amount in the bloodstream is modelled by
D(t) = 200 · (0.7)t, t in hours
Set up: What are we solving for?
(i) Sketch D(t) for 0 ≤ t ≤ 8, showing the asymptote and y-intercept. 2 marks
(ii) Compute D(2), D(4), D(8) to the nearest milligram. By what factor does the dose decrease each hour, and by what factor over a 2-hour gap? 3 marks
(iii) A clinician says a "therapeutic threshold" is 50 mg. Use your sketch (or trial values) to estimate the largest whole number of hours t for which D(t) ≥ 50 mg. 2 marks
Stuck on (iii)? Calculate D(3), D(4), D(5) and find when it drops below 50.Problem 4 — Pure transformations
Use the base graph y = 2x to describe each transformed graph below.
Set up: What are we solving for?
(i) Describe (in words, in order) the sequence of transformations that maps y = 2x onto y = −2x + 1 + 3. 3 marks
(ii) State the asymptote, y-intercept and range of y = −2x + 1 + 3. 3 marks
(iii) Explain in one sentence why the reflection in the x-axis turns "growth" behaviour (y > 0 for all x) into y < 0 for all x. 1 mark
Problem 5 — Comparing growth and decay graphs
The graphs of y = 2x and y = 2−x are drawn on the same axes.
Set up: What are we solving for?
(i) Show algebraically that y = 2−x is identical to y = (1/2)x. 1 mark
(ii) State the geometric relationship between the two graphs, and explain why this is so (one sentence using "reflection"). 2 marks
(iii) Find their single point of intersection by setting 2x = 2−x. Show one line of working. 2 marks
Stuck on (iii)? Equal bases ⇒ equal exponents: x = −x.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Cooling coffee
Set up. Read the asymptote, compute initial and intermediate values, sketch and interpret the long-run behaviour.
(i) Asymptote: T = 20°C. Physically, this is the room (ambient) temperature; the coffee cools toward room temperature, never below it.
(ii) T(0) = 70 · 1 + 20 = 90°C (the initial temperature). T(5) = 70 · (0.9)5 + 20 = 70 · 0.59049 + 20 ≈ 41.3 + 20 = 61.3°C.
(iii) Sketch: dashed asymptote T = 20, y-intercept (0, 90), curve decreasing through (5, 61.3) and approaching T = 20 from above. Long-run temperature: 20°C (the asymptote).
Problem 2 — Fish population
Set up. Fit y = a · bx to two data points to find both parameters.
(i) At (0, 5): 5 = a · b0 = a, so a = 5.
(ii) At (2, 20): 20 = 5 · b2 ⇒ b2 = 4 ⇒ b = 2 (positive root). Model: y = 5 · 2x.
(iii) y(5) = 5 · 25 = 5 · 32 = 160 (thousand fish). Assumption: the lake has sufficient food/space for the exponential trend to continue — in reality the population will be limited by carrying capacity and the model will overpredict for large x.
Problem 3 — Drug clearance
Set up. Sketch and read off decay values from D(t) = 200 · (0.7)t.
(i) Sketch: asymptote D = 0, y-intercept (0, 200), decreasing curve approaching the t-axis.
(ii) D(2) = 200 · 0.49 = 98 mg. D(4) = 200 · 0.2401 ≈ 48 mg. D(8) = 200 · 0.05765 ≈ 12 mg. Per hour the dose is multiplied by 0.7 (a 30% drop each hour). Over a 2-hour gap, by 0.72 = 0.49 (a 51% drop).
(iii) D(3) = 200 · 0.343 ≈ 69 mg (≥ 50). D(4) ≈ 48 mg (< 50). Largest whole-number t with D(t) ≥ 50 is t = 3 hours.
Problem 4 — Transformations
Set up. Decompose a multi-transform expression into ordered single transformations and read off graph features.
(i) In order: (1) shift left 1 (x + 1 inside the exponent); (2) reflect in the x-axis (the negative sign in front); (3) shift up 3.
(ii) Asymptote: y = 3 (from the +3 outside). y-int at x = 0: y = −21 + 3 = −2 + 3 = 1; point (0, 1). Range: y < 3 (the reflection sends the original y > 0 to y < 0; adding 3 shifts to y < 3).
(iii) Reflecting in the x-axis multiplies every y-value by −1; since y = 2x > 0 for all x, the reflected y = −2x < 0 for all x.
Problem 5 — Growth vs decay graphs
Set up. Show two equivalent forms, describe the geometric symmetry, and find the intersection algebraically.
(i) 2−x = (2−1)x = (1/2)x. ✓
(ii) The graphs are reflections of each other in the y-axis. Replacing x by −x reflects every point (x, y) to (−x, y).
(iii) 2x = 2−x ⇒ x = −x ⇒ 2x = 0 ⇒ x = 0. At x = 0: y = 20 = 1. Intersection: (0, 1) — the common y-intercept.