Unit 6

Introduction to Exponential Functions

Unit Narrative

In this unit, students build on their understanding of linear functions, properties of exponents, and percent change to explore exponential relationships. Students learn that exponential relationships are characterized by a constant quotient over equal intervals, and compare it to linear relationships, which are characterized by a constant difference over equal intervals. They encounter contexts that change exponentially. These contexts are presented verbally and with tables and graphs. They construct equations and use them to model situations and solve problems. At first, students investigate these exponential relationships without using function notation...

to access Pre-Unit Assessments.

Section A

Section Goals

Section Narrative

In this section, students begin their exploration of exponential functions. They consider patterns of numbers that increase by a common factor and compare them to patterns that increase by a common difference. This motivates the need for a way to describe these non-linear patterns with exponential expressions.

number of errands	pay in dollars	difference in pay from previous week	factor...

Let's choose the better deal.

Let’s compare different patterns of growth.

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Section A

Let's choose the better deal.

Let’s compare different patterns of growth.

Section B

Section Goals

Section Narrative

In this section, students write expressions to represent exponential growth. They make use of properties of exponents and repeated multiplication of a growth factor. They are introduced to patterns that decrease in an exponential way, by looking at growth factors between 0 and 1 and by rewriting expressions using the distributive property.

Later in the section, students examine graphs of...

Let’s explore exponential growth.

Let’s think about how to show and talk about exponential decay.

Let’s look at exponential decay.

Let's compare situations in which quantities change exponentially.

Let’s look more closely at exponential graphs and equations.

to access Section Checkpoints.

Section B

Let’s explore exponential growth.

Let’s think about how to show and talk about exponential decay.

Let’s look at exponential decay.

Let's compare situations in which quantities change exponentially.

Let’s look more closely at exponential graphs and equations.

Section C

Section Goals

Section Narrative

In this section, students begin writing exponential relationships as functions. They revisit function notation and apply language associated with functions, such as dependent and independent variables. They also consider average rate of change in an exponential context and conclude that these measures are less representative of the function than when using linear functions.

Later, students examine the impact of parameters...

Let’s explore exponential functions.

Let’s find some meaningful ways to represent exponential functions.

Let's calculate average rates of change for exponential functions.

Let’s use exponential functions to model real-life situations.

Let’s study and compare equations and graphs of exponential functions.

Let’s investigate what we can learn from graphs that represent exponential functions.

to access Section Checkpoints.

to access Mid-Unit Assessments.

Section C

Let’s explore exponential functions.

Let’s find some meaningful ways to represent exponential functions.

Let's calculate average rates of change for exponential functions.

Let’s use exponential functions to model real-life situations.

Let’s study and compare equations and graphs of exponential functions.

Let’s investigate what we can learn from graphs that represent exponential functions.

Section D

Section Goals

Section Narrative

In this section, students apply their understanding of exponential change to situations involving percent increase and decrease. In particular, students explore the context of compounding interest in depth including application of interest at different frequencies.

To simplify the math, many of the situations are not realistic. An initial amount of money is deposited and left untouched for years or a...

Let's find the result of changing a number by a percentage.

Let's investigate what happens when we repeatedly apply a percent increase to a quantity.

Let's explore different ways of repeatedly applying a percent increase.

Let's find out what happens when we repeatedly apply the same percent increase at different intervals of time.

Let's write exponential expressions in different ways.

to access Section Checkpoints.

Section D

Let's find the result of changing a number by a percentage.

Let's investigate what happens when we repeatedly apply a percent increase to a quantity.

Let's explore different ways of repeatedly applying a percent increase.

Let's find out what happens when we repeatedly apply the same percent increase at different intervals of time.

Let's write exponential expressions in different ways.

Section E

Section Goals

Section Narrative

In this section, students revisit comparisons between linear and exponential change, now with the focus on functions. They see that exponential functions eventually grow more quickly than do linear functions. They also justify that functions of the form have a constant difference of between any 2 inputs that are 1 unit apart, while functions of the form have a constant...

Let's compare linear and exponential functions as they continue to increase.

Let's explore how linear and exponential functions change over equal intervals.

to access Section Checkpoints.

Section E

Let's compare linear and exponential functions as they continue to increase.

Let's explore how linear and exponential functions change over equal intervals.

Section F

Section Narrative

In this final section, students have the opportunity to apply their thinking from throughout the unit. As this is a short section followed by an End-of-Unit Assessment, there are no section goals or checkpoint questions.

Let's use linear and exponential models to represent and understand population changes.

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Section F

Let's use linear and exponential models to represent and understand population changes.

to access End-of-Unit Assessments.

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Introduction to Exponential Functions

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