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

Sketch the graph of the exponential function $f(x)=3\cdot2^{x-1}-4$. On your sketch, clearly indicate the horizontal asymptote and the x- and y-intercepts.

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

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

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

Find and sketch the exponential model of the form $y=A\cdot3^{x}+C$ that passes through the points $(0,5)$ and $(1,14)$. Indicate the asymptote and intercepts.

Given the graph of an exponential function with horizontal asymptote $y=2$ and passing through $(0,5)$ and $(2,10)$, find its equation in the form $f(x)=A\cdot b^x+2$ and sketch the graph.

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.

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