options: For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Distinguish between situations that can be modeled with linear functions and with exponential functions.

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
problem_activity: Activity
In a previous lesson, students used equations defining functions to reason about graphs. Here they work in the opposite direction, analyzing graphs and creating corresponding equations. In the first question, the coordinates given are not of two consecutive values of
\(x\)
, so students need to reason about the decay factor indirectly. They may:
Try different factors that, when multiplied by
\(f(0)\)
twice, give
\(f(2)\)
.
Use the graph to estimate
\(f(1)\)
, and then use that estimate to find a reasonable decay factor.
Find the factor that takes
\(f(0)\)
to
\(f(2)\)
(or
\(f(2)\)
to
\(f(4)\)
), recognize that number as
\(b^2\)
, and then find
\(b\)
.
The second question has no context and is thus more abstract. This prepares students to interpret graphs in the next activity, where minimal information is given.
Launch
Arrange students in groups of 2. Ask students to take a few minutes of quiet think time on each question before discussing their thinking with a partner.
Reading, Conversing, Writing: MLR5 Co-Craft Questions.
To help students make sense of an exponential situation in which the graph is given before the equation, start by displaying only the first task statement and its graph. Give students 1–2 minutes to write their own mathematical questions about the situation. Invite students to share their questions with the class, then reveal the questions. Listen for and amplify any questions seeking to write an equation, interpret the meaning of various points, or predict future trends. This will help students create questions and use language that demonstrates what they know is true about a situation given a graph first.
Design Principle(s): Maximize meta-awareness; Cultivate conversation
Student Facing
Here is a graph representing an exponential function
\(f\)
. The function
\(f\)
gives the value of a computer, in dollars, as a function of time,
\(x\)
, measured in years since the time of purchase.
[IMG]
Based on the graph, what can you say about the following?
The purchase price of the computer
The value of
\(f\)
when
\(x\)
is 1
The meaning of
\(f(1)\)
How the value of the computer is changing each year
An equation that defines
\(f\)
Whether the value of
\(f\)
will reach 0 after 10 years
Here are graphs of two exponential functions. For each, write an equation that defines the function and find the value of the function when
\(x\)
is 5.
[IMG]
[IMG]
Student Response
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Student Facing
Are you ready for more?
Consider a function
\(f\)
defined by
\(f(x)=a\boldcdot b^x\)
.
If the graph of
\(f\)
goes through the points
\((2,10)\)
and
\((8,30)\)
, would you expect
\(f(5)\)
to be less than, equal to, or greater than 20?
If the graph of
\(f\)
goes through the points
\((2,30)\)
and
\((8,10)\)
, would you expect
\(f(5)\)
to be less than, equal to, or greater than 20?
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
If students have trouble getting started, ask them to consider the graph and say what they know is true about the situation that the graph represents. Most likely, they will answer some parts of the first question by doing this, and will be in a better position to figure out any questions they didn't answer.
Activity Synthesis
Select students to share their responses to the first question. Focus the discussion on how students found the decay factor for the computer. Make sure students understand that the key is to notice that every 2 years, the computer's value is multiplied by
\(\frac{1}{4}\)
. This means that the annual decay factor is
\(\frac{1}{2}\)
, since
\(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\)
.
For the last graph, discuss questions such as:
“What information is needed to describe an exponential function?” (the factor of growth or decay and the initial value)
“Where on the graph do we see the initial value?” (the
\(y\)
-intercept of the graph)
“What points on the graph can we use to help us find the growth factor?” (the points
\((\text-1, 40)\)
and
\((0,20)\)
, or
\((0,20)\)
and
\((2,5)\)
)
“Why might
\((\text-1,40)\)
and
\((0,20)\)
be a more strategic choice than
\((0,20)\)
and
\((2,5)\)
for finding the factor of growth?” (The
\(x\)
coordinates of first two points differ by 1, so the quotient of the
\(y\)
coordinates of those points can be used to find the factor of decay. It is a more direct way to find the factor of decay, which is the factor by which the output of
\(f\)
changes when the input
\(x\)
changes by 1. In the second pair, the
\(x\)
coordinate changes by 2.)
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, display the equation
\(100 = 400\boldcdot (b)^2\)
and use color or annotations to illustrate how each value or variable is represented graphically.
Supports accessibility for: Visual-spatial processing; Conceptual processing


correct_option_index: 3