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.