Exercises for Question Type 13: Formulating and plotting exponential models to indicate key features such as asymptotes and intercepts

Question 1

Mathematical modeling of population growth using exponential functions, including sketching graphs and identifying key features such as asymptotes, intercepts, and specific values.

A population of fish is modelled by P (t) = 50 e^{0.08 t}, where t is the time in years. Sketch the graph of P (t) for 0 \leq t \leq 50 . Indicate the horizontal asymptote, the initial population, and the population at t = 50 .

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Question 2

Sketch the graph of the exponential function f (x) = 3 \cdot 2^{x - 1} - 4 . On your sketch, clearly indicate the horizontal asymptote and the x- and y-intercepts.

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Question 3

Fit an exponential model y = A e^{k x} to the data points (1, 5) and (4, 40) . Then sketch the graph, indicating the asymptote and intercept.

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Question 4

Questions based on exponential decay models and their graphical representations.

The temperature of a cooling object is modelled by T (t) = 20 + 80 e^{- 0.15 t} where t is the time in minutes and T is the temperature in degrees Celsius. Sketch the graph of T for t \geq 0, stating the horizontal asymptote and the initial temperature.

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Question 5

Find and sketch the exponential model of the form y = A \cdot 3^{x} + C that passes through the points (0, 5) and (1, 14) . Indicate the asymptote and intercepts.

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Question 6

The half-life of a radioactive isotope is 5 days, and its initial mass is 100 g. The mass, M grams, of the isotope after t days can be modelled by the function: M (t) = 100 \times 2^{- t /5}, for t \geq 0

(a)

State the equation of the horizontal asymptote of the graph of M .

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(b)

Calculate the mass of the isotope when t = 20 .

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(c)

Sketch the graph of M for 0 \leq t \leq 20 . Clearly show the horizontal asymptote, the initial mass, and the point where t = 20 .

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Question 7

A bank account grows continuously at 5% annual interest. If A (t) = 1000 e^{0.05 t}, sketch the graph of A (t), indicating the horizontal asymptote, the initial amount, and the amount after 20 years.

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Question 8

Determine and sketch the exponential function of the form f (x) = A \cdot b^{x} + D given that its horizontal asymptote is y = 3, it passes through (0, 7) and its growth factor is 1.5 per unit increase in x . Indicate intercepts.

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Question 9

A medication concentration in blood follows C (t) = C_{0} e^{- 0.25 t} . If the initial concentration is 200 mg/L, sketch C (t), specify the horizontal asymptote, and compute the concentration after 10 hours.

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Question 10

The graph of an exponential function has a horizontal asymptote y = 2 and passes through (0, 5) and (2, 10) . Find its equation in the form f (x) = A \cdot b^{x} + 2 and sketch the graph.

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Question 11

An aquarium’s pH level increases following the model pH (t) = 6 + 2 (1 - e^{- 0.3 t}) . Graph pH (t) for 0 \leq t \leq 20, stating the horizontal asymptote and initial value.

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IB Mathematics Applications & Interpretation Question Type 13: Formulating and Plotting Exponential Models to Indicate Key Features Such as Asymptotes and Intercepts Exercises