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

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

Written by official IB examiners

Question 1

Skill question

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 $t = 50$ .

[5]

Question 2

Skill question

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.

[4]

Question 3

Skill question

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.

[9]

Question 4

Skill question

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.

[5]

Question 5

Skill question

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.

[8]

Question 6

Skill question

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 $M$ .

[1]

(b) Calculate the mass of the isotope when $t = 20$ .

[2]

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

[3]

Question 7

Skill question

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.

[6]

Question 8

Skill question

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.

[5]

Question 9

Skill question

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.

[6]

Question 10

Skill question

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.

[6]

Question 11

Skill question

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.

[5]

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

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