# Exponential Functions 2

## •  Create and analyze a simple exponential function arising from a real-world or mathematical context (F-LE.A.2$^\star$). •  Evaluate and interpret exponential functions at non-integer inputs (N-RN.A.1, F-LE.A.2$^\star$). •  Understand functions of the form $f(t) = P(1 + r/n)^{nt}$ and solve problems with different compounding intervals (A-SSE.A.1$^\star$, F-LE.A.2$^\star$). •  Understand informally how the base $e$ is used in functions to model a quantity that compounds continuously (F-BF.A.1a$^\star$). •  Write exponential expressions in different forms (F-LE.A.2$^\star$, F-BF.A.1$^\star$). •  Explain what the parameters of an exponential function mean in different contexts (F-LE.B.5$^\star$). •  Use the properties of exponents to write expressions in equivalent forms (A-SSE.B.3a$^\star$). •  Build exponential functions to model real world contexts (F-LE.A.1$^\star$, F-LE.A.2$^\star$, F-BF.A.1$^\star$). •  Analyze situations that involve geometric sequences and series (A-SSE.A.1$^\star$). •  Derive the formula for the sum of a finite geometric series (A-SSE.B.4$^\star$).

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 $ab^x = c$ 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

A2.2.0 Pre-unit diagnostic assessment

#### Summary

Diagnose students’ ability to
• create basic exponential functions (F-LE.A.2$^\star$);
• create an exponential model for a simple context (F-BF.A.1$^\star$);
• apply exponent rules (N-RN.A.1).

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A2.2.1 Understanding exponential growth and decay

#### Summary

Create and analyze a simple exponential function arising from a real-world or mathematical context (F-lE.A.2$^\star$).

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A2.2.2 Exponential functions on the real numbers

#### Summary

Evaluate and interpret exponential functions at non-integer inputs(N-RN.A.1, F-LE.A.2$^\star$).

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A2.2.3 Changing compounding intervals, continuous compounding, and the base e

#### Summary

• Understand functions of the form $f(t) = P(1 + r/n)^{nt}$ which use a given compounding frequency, $n$ (A-SSE.A.1$^\star$).
• Solve problems given different compounding intervals (F-LE.A.2$^\star$).
• Understand (informally) how the base $e$ arises in the context of compounding intervals as $n$ becomes arbitrarily large (F-BF.A.1a$^\star$).

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A2.2.4 Interpreting exponential functions

#### Summary

• Solve problems involving exponential functions in many different contexts (F-LE.A.2$^\star$).
• Write exponential expressions in different forms (F-BF.A.1$^\star$).
• Explain what the parameters of an exponential function mean in different contexts (F-LE.B.5$^\star$).
• Use the properties of exponents to write expressions in equivalent forms (A-SSE.B.3c$^\star$).

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A2.2.5 Modeling with exponential functions

#### Summary

Build exponential functions to model real world contexts (F-LE.A.1$^\star$, F-LE.A.2$^\star$, F-BF.A.1$^\star$).

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A2.2.6 Geometric sequences and series

#### Summary

• Analyze situations that involve geometric sequences and series (A-SSE.A.1$^\star$).
• Derive the formula for the sum of a finite geometric series (A-SSE.B.4$^\star$).

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A2.2.7 Summative assessment

#### Summary

Assess students' ability to
• evaluate and interpret exponential functions at non-integer inputs (N-RN.A.1, F-LE.A.2$^\star$);
• construct exponential functions given a description or data from a table (F-LE.A.2$^\star$);
• manipulate exponential expressions (A-SSE.B.3$^\star$);
• use the formula for the sum of a finite geometric series to solve a problem (A-SSE.B.4$^\star$);
• create and analyze simple exponential functions arising from real-word data (F-LE.B.5$^\star$).

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