Reasoning About Exponential Graphs

Which equation matches this graph?

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Which equation matches this graph?

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Which graph matches this equation?

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The revenues of two companies can be modeled with exponential functions \(f\) and \(g\). Here are the graphs of the two functions. In each function, the revenue is in thousands of dollars and time, \(t\), is measured in years. The \(y\)-coordinate of the intersection is 215.7. Select all statements that correctly describe what the two graphs reveal about the revenues.

• If we know at least one of the expressions that define \(f\) and \(g\), we can calculate the value of \(t\) at the intersection.

• At the intersection, \(f(t) = 215.7\) and \(g(t) = 215.7\).

• The intersection of the graphs tells us when the revenues of the two companies grow by the same factor.

• The intersection tells us when the two companies have the same revenue.

• At the intersection, \(f(t) > g(t)\).

The population of a fast-growing city in Texas can be modeled with the equation \(p(t) = 82 \cdot e^{(0.078t)}\). The population of a fast-growing city in Tennessee can be modeled with \(q(t) = 132 \cdot e^{(0.047t)}\). In both equations, \(t\) represents years since 2016 and the population is measured in thousands. The graphs representing the two functions are shown. The point where the two graphs intersect has a \(y\)-coordinate of about 271.7.

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Here are graphs of the functions \(f\) and \(g\) given by \(f(x) = 100 \cdot (1.2)^x\) and \(g(x) = 100 \cdot e^{0.2x}\). Which graph corresponds to each function? Explain how you know.

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Here is a graph that represents \(f(x) = e^x\). Explain how we can use the graph to estimate: The solution to an equation such as \(300 = e^x\). The value of \(\ln 700\).

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The function \(f\) is given by \(f(x) = 100 \cdot 3^x\). Select all equations whose graph meets the graph of \(f\) for a positive value of \(x\).

• \(y = 10 \cdot e^x\)

• \(y = 500 \cdot e^x\)

• \(y = 500 \cdot e^{-x}\)

• \(y = 1,000 \cdot 2^x\)

• \(y = 600 \cdot 10^x\)