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

A population of fish is modelled by $P(t) = 50e^{0.08t}$, where $t$ is the time in years.

Sketch the graph of $P(t)$ for $0 ≤ t ≤ 50$. Indicate the horizontal asymptote, the initial population, and the population at $t = 50$.

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.

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

The temperature of a cooling object is modelled by $T(t) = 20 + 80e^{-0.15t}$ where $t$ is the time in minutes and $T$ is the temperature in degrees Celsius.

Sketch the graph of $T$ for $t \ge 0$, stating the horizontal asymptote and the initial temperature.

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.

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

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.

A medication concentration in blood follows $C(t)=C_0e^{-0.25t}$. If the initial concentration is $200 \text{ mg/L}$, sketch $C(t)$, specify the horizontal asymptote, and compute the concentration after 10 hours.

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.

An aquarium’s pH level increases following the model $\text{pH}(t)=6+2\bigl(1-e^{-0.3t}\bigr)$. Graph $\text{pH}(t)$ for $0\le t\le20$, stating the horizontal asymptote and initial value.