Grade/Course: Algebra I (Second Semester)
Instructional Unit 7: Understanding Exponential Functions, Models, and Equations
Instructional Schedule: Third Nine Weeks (suggested for 20 days)
Adapted from Timothy Kanold Scope-and-Sequence documents
Standards:
Prerequisite Knowledge:
Assessment Tools:
(standards linked to content taught
(formative assessments, quizzes,
in previous grades)
mastery tasks/activities)
Construct and compare linear, quadratic, and exponential models and solve problems. (supporting content)
(BA/PASS 2.2e, 2.5a) Construct linear
and exponential functions (from a
table, graph, or other situation),
including arithmetic and geometric
sequences, given a graph, a
description of a relationship, or two
input-output pairs (including reading
these from a table).
(BA 2.5) Prove that linear functions
grow by equal differences over equal
intervals, and that exponential
functions grow by equal factors over
equal intervals.
(BA/PASS 2.2e) Recognize situations
in which one quantity changes at a
constant rate per unit interval
relative to another.
(BA/PASS 2.5a) Recognize situations
in which a quantity grows or decays
by a constant percent rate per unity
interval relative to another.
Evidence Of Standard:
(student should be able to…)
-Explain why a function is linear or
exponential from data presented in a
graph, a table, or a written
description.
-Construct linear and/or exponential
functions given a graph, a description
of a relationship, or two input-output
pairs (including reading those from a
table).
-Distinguish between linear functions
and exponential functions.
-Demonstrate and prove that a linear
function has a constant slope (rate of
change) over equal intervals.
-Demonstrate and prove that an
exponential function grows at a
constant multiplier ( 21 , 22 , 23 , 𝑒𝑡𝑐)
over equal intervals.
-Recognize real-world, as well as
theoretical, situations in which a
quantity changes at a constant rate
per unit interval relative to another.
-Recognize real-world, as well as
theoretical, situations in which a
quantity grows or decays by a
constant percent rate per unit
interval relative to another.
(BA 2.5d) Observe using graphs and
tables that a quantity increasing
exponentially eventually exceeds a
quantity increasing linearly,
quadratically, or (more generally) as
a polynomial function.
-Use graphs and tables to show and
compare the different output values
and rates of change for linear and
exponential functions.
-Understand that a quantity
increasing exponentially eventually
exceeds a quantity increasing linearly
due to the multiple factor.
Interpret the parameters in a linear or exponential function in terms of a context. (supporting content)
(BA/PASS 3.1b) Interpret the
parameters in a linear or exponential
function in terms of a context which
could lead to inferences or
predictions based on data from
graphs, tables, and charts.
-Interpret and understand that
quantities, rates of change, and other
values of a linear function
f(x) = mx + b in the context of a realworld situation.
- Interpret and understand that
quantities, rates of change, and other
values of an exponential function
f(x) = 𝑎𝑏 𝑥 + c in the context of a realworld situation.
Build a function that models a relationship between two quantities. (supporting content)
(BA 2.1f) Determine an explicit
expression, a recursive process, or
steps for calculation from a context.
(BA 2.1g) Combine standard function
types using arithmetic operations.
For example, build a function that
models the temperature of a cooling
body by adding a constant function
to a decaying exponential.
-Distinguish between an explicit and
recursive expression of a function.
-Write an explicit expression of a
function to describe a real-world
scenario.
-Write a recursive expression of a
function to describe a real-world
scenario.
-Determine the steps for calculation
for a real-world scenario.
-Combine standard function types
using arithmetic operations for a realworld scenario.
Build new functions from existing functions. (additional content)
(BA/PASS 2.2b) Identify the effect on
the graph of an exponential function
by replacing 𝑓(𝑥) by 𝑓(𝑥) + 𝑘,
𝑘 𝑓(𝑥), 𝑓(𝑘𝑥), and 𝑓(𝑥 + 𝑘) for
specific values of 𝑘 (both positive and
negative); find the value 𝑘 given the
graphs. Experiment with cases and
illustrate an explanation of the
effects on the graph using
technology. Include recognizing even
and odd functions from their graphs
and algebraic expression for them.
-Identify and explain the effect of
basic transformations on the parent
function of an exponential function
including:
1. 𝑓(𝑥) + 𝑘
2. 𝑘 𝑓(𝑥)
3. 𝑓(𝑘𝑥)
4. 𝑓(𝑥 + 𝑘)
-Find the specific value of k (both
positive and negative) given a graph
of the function.
Analyze functions using different representations. (supporting content)
(BA/PASS 2.2a, 2.5a) Graph functions
expressed symbolically and show key
features of the graph, by hand in
simple cases and using technology for
more complicated (exponential
only).
(BA 2.6) Write a function defined by
an expression in different but
equivalent forms to reveal and
explain different properties of a
function.
(BA/PASS 2.1d) Compare properties
of two functions each represented in
a different way (algebraically,
graphically, numerically in tables, or
by verbal description). For example,
given a graph of one exponential
function and an algebraic expression
for another, say which has the larger
maximum.
-Graph exponential functions and
identify key features of the function
from the graph.
-Classify exponential models that
represent exponential growth and
exponential decay.
-Use properties of exponents to
interpret expressions for exponential
functions.
--Compare properties of two
functions represented in different
ways (algebraically, graphically,
numerically in tables, or by verbal
descriptions).
Interpret functions that arise in applications in terms of the context. (key content)
(BA/PASS 2.2d) For a function that
models a relationship between two
quantities, interpret key features of
graphs and tables in terms of the
quantities, and sketch graphs
showing key features given a verbal
description of the relationship. Key
features include: intercepts; intervals
where the function is increasing,
decreasing, positive, or negative;
relative maximums and minimums;
symmetries; end behavior, and
periodicity.
-Understand how relationships
between two quantities are conveyed
through:
1. x and y intercepts
2. ordered pairs
3. increasing intervals
4. decreasing intervals
5. positive intervals
6. negative intervals
7. Symmetries
8. Ordered pairs
-Recognize key information in written
problems as components of an
underlying function and sketch a
graph that conveys this information
and indicates all key features of the
underlying function.
(BA/PASS 2.2c.1) Calculate and
-Calculate and interpret the average
interpret the average rate of change
rate of change of a function
of a function (presented symbolically presented symbolically over a
or as a table) over a specified
specified interval.
interval. Estimate the rate of change -Calculate and interpret the average
from a graph. (Percent rate of
rate of change of a function
change)
presented in a table over a specified
interval.
-Calculate and interpret the average
rate of change of a function
presented in function notation over a
specified interval.
-Estimate the rate of change from a
graph.
Write expressions in equivalent forms to solve problems. (supporting content)
(BA/PASS 1.1d) Use the properties of
exponents to transform expressions
for exponential functions. For
example, the expression 1.15𝑡 can be
12𝑡
- Use the properties of exponents to
transform expressions for
exponential functions. For example,
the expression 1.15𝑡 can be written
12𝑡
written as (1.151/12 ) = 1.01212𝑡
as (1.151/12 ) = 1.01212𝑡 to
to reveal the approximate equivalent reveal the approximate equivalent
monthly interest rate if the annual
monthly interest rate if the annual
rate is 15%.
rate is 15%.
Summarize, represent, and interpret data on two categorical and quantitative variables. (supporting content)
(BA/PASS 3.2a) Fit a function to the
data; use functions fitted to data to
solve problems in the context of the
data. Use given functions or choose
a function suggested by the context.
Emphasize liner and exponential
models.
-Decide which type of function
(linear, quadratic, exponential) is
appropriate to represent a data set.
-Solve problems by using a function
appropriately fitted to the data set.
-Analyze the scale and shape of a
scatter plot to estimate the function
with the best fit for the data set.
(BA/PASS 3.2b) Informally assess the -Calculate and plot residuals for the
fit of a function by plotting and
data set and function with a possible
analyzing residuals.
fit.
-Informally assess the fit of a function
by analyzing residuals.
Note: Any italicized text denotes portions of a given standard that do not apply to identified standard content in this unit.
Resources/Exemplar Tasks:
( list possible task/activities students could engage in within this unit)
Standards for Mathematical Practice:
(highlight practice standards to be emphasized in the instructional unit)
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of instruction.
8. Look for and express regularity in repeated reasoning.
( BA: Broken Arrow rigor standard; PASS: Priority Academic Student Skills standard; BA/PASS: Combination standard )