# Exponential Functions 2

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• Create and analyze a simple exponential function arising from a real-world or mathematical context.

• Evaluate and interpret exponential functions at non-integer inputs.

• Understand functions of the form $f(t) = P(1 + r/n)^{nt}$ and solve problems with different compounding intervals.

• Understand informally how the base $e$ is used in functions to model a quantity that compounds continuously.

• Write exponential expressions in different forms.

• Explain what the parameters of an exponential function mean in different contexts.

• Use the properties of exponents to write expressions in equivalent forms.

• Build exponential functions to model real world contexts.

• Analyze situations that involve geometric sequences and series.

• Derive the formula for the sum of a finite geometric series.

In previous units students have worked with geometric sequences and understand they change by a constant ratio over a constant interval. They are able to write both recursive and closed equations for them. Students understand the difference between a linear and exponential function, can recognize situations and tables described by each, and know that an exponential function will always overtake a linear function. They know that an exponential function grows increasingly rapidly in one direction, and approaches a value asymptotically in the other direction. They have solved exponential equations of the form abx = c by graphing. They can construct an exponential function given a graph, description of a relationship, or two input output pairs with integer inputs (including in a table). Given an expression defining an exponential function, they can interpret its parameters in a context. They can also fit a simple exponential function to a scatterplot. Every exponential function until now has only involved integer inputs.

In this unit students broaden their view of exponential functions to include the entire real number line as a possible domain. They learn about functions with base $e$. An approach using compound interest that shows e arising as the natural base for a quantity being compounded continuously can serve as a way to develop understanding appropriate to this level. First, students must understand functions of the form $f(x) = P(1 + r/n)^{nt}$ which show a given compounding frequency, $n$.

Students examine some different forms of exponential functions and learn to interpret the parameters in terms of a context. They learn the concept of doubling time and see functions expressed in a form that shows the doubling time; they work algebraically with functions expressed in a form like $f(x) = A(1 + r/n)^{nt}$ that shows the compounding period; and they work with functions written with the base e, $g(x) = Ae^{rt}$, in many continuous growth contexts. Students build functions in those forms in order model real-world contexts. Contexts may include Moore’s law for computer processor speeds, population growth, and temperature change. Students should also consider whether an exponential or a linear model is appropriate in various contexts.

Students analyze situations that involve summing an exponential sequence (which is generally called a geometric series) These arise naturally in some saving and banking problems as well as in some interesting geometric contexts. Students will build on previous work with geometric sequences to derive a formula for the sum of a geometric series.

A later unit introduces logarithms as the solutions to exponential equations. Ultimately, students are proficient at graphing, analyzing, solving, and modeling exponential situations. Students are comfortable with exponentials from a functions viewpoint. This lays the foundations for success in calculus.

## Sections

#### Summary

Create and analyze a simple exponential function arising from a real-world or mathematical context.

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#### Summary

• Understand functions of the form $f(t) = P(1 + r/n)^{nt}$ which use a given compounding frequency, $n$.

• Solve problems given different compounding intervals.

• Understand (informally) how the base $e$ arises in the context of compounding intervals as $n$ becomes arbitrarily large.

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#### Summary

• Solve problems involving exponential functions in many different contexts.

• Write exponential expressions in different forms.

• Explain what the parameters of an exponential function mean in different contexts.

• Use the properties of exponents to write expressions in equivalent forms.

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#### Summary

• Analyze situations that involve geometric sequences and series A-SSE.

• Derive the formula for the sum of a finite geometric series.

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