In Module 3, Topic C, students graph logarithmic functions, identifying key features (F-IF.4, F-IF.7) and discover how the logarithmic properties are evidenced in the graphs of corresponding logarithmic functions. The inverse relationship between an exponential function and its corresponding logarithmic function is made explicit (F-BF.3). In the final lesson in Topic C, students synthesize what they know about linear, quadratic, sinusoidal, and exponential functions to determine which function is most appropriate to use to model a variety of real-world scenarios (F-BF.1a).

Content Standard(s):

Mathematics MA2015 (2016) Grade: 9-12 Algebra II

35 ) Find inverse functions. [F-BF4]

a. Solve an equation of the form f(x) = c for a simple function f that has an inverse, and write an expression for the inverse. [F-BF4a]

Example: f(x) =2x^{3} or f(x) = (x+1)/(x-1) for x ≠ 1.

Mathematics MA2015 (2016) Grade: 9-12 Algebra II

29 ) Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [F-IF5]

Example: If the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

Mathematics MA2015 (2016) Grade: 9-12 Algebra II

36 ) 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. [F-LE4]

Mathematics MA2019 (2019) Grade: 9-12 Algebra I with Probability

17. Combine different types of standard functions to write, evaluate, and interpret functions in context. Limit to linear, quadratic, exponential, and absolute value functions.

a. Use arithmetic operations to combine different types of standard functions to write and evaluate functions.

Example: Given two functions, one representing flow rate of water and the other representing evaporation of that water, combine the two functions to determine the amount of water in a container at a given time.

b. Use function composition to combine different types of standard functions to write and evaluate functions.

Example: Given the following relationships, determine what the expression S(T(t)) represents.

Function

Input

Output

G

Amount of studying: s

Grade in course: G(s)

S

Grade in course: g

Amount of screen time: S(g)

T

Amount of screen time: t

Number of follers: T(t)

Unpacked Content

Evidence Of Student Attainment:

Students: Given different types of standard functions

Use arithmetic operations to combine functions in context.

Use function composition to combine functions in context.

Write, evaluate, and interpret combined functions in context.

Teacher Vocabulary:

Function composition

Knowledge:

Students know:

Techniques to combine functions using arithmetic operations.

Techniques for combining functions using function composition.

Skills:

Students are able to:

Accurately develop a model that shows the functional relationship between two quantities.

Accurately create a new function through arithmetic operations of other functions.

Present an argument to show how the function models the relationship between the quantities.

Understanding:

Students understand that:

Arithmetic combinations of functions may be used to improve the fit of a model.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: ALGI.17.1: Define functions, relations (ordered pairs), input, output.
ALGI.17.2: Recall how to complete input/output tables.
ALGI.17.3: Recall how to read/interpret information from a table.
ALGI.17.4: Identify algebraic expressions.
ALGI.17.5: Recall how to name points from a graph (ordered pairs).
ALGI.17.6: Recall how to name points on a Cartesian plane using ordered pairs.

a.
ALGI.17.7: Identify, represent, and analyze two quantities that change in relationship to one another in real-world or mathematical situations.
ALGI.17.8: Set up an equation to represent the given situation, using correct mathematical operations and variables.

b.
ALGI.17.9: Add, subtract, and multiply polynomials, showing that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtration, and multiplication.

Prior Knowledge Skills:

Explain the distributive property.

Give examples of the properties of operations including distributive.

Combine like terms of a given expression.

Recognize the correct order to solve expressions with more than one operation.

Calculate a numerical expression (Ex. V=(4x4x4).

Choose the correct value to replace each variable in the algebraic expression (Substitution).

Calculate an expression in the correct order (Ex. exponents, mult./div. from left to right, and add/sub. from left to right).

Mathematics MA2019 (2019) Grade: 9-12 Algebra I with Probability

23. Identify the effect on the graph of replacing f(x) by f(x)+k,k·f(x), f(k·x), and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and explain the effects on the graph, using technology as appropriate. Limit to linear, quadratic, exponential, absolute value, and linear piecewise functions.

Unpacked Content

Evidence Of Student Attainment:

Students:
Given a function in algebraic form,

Graph the function, f(x), conjecture how the graph of f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k(both positive and negative) will change from f(x), and test the conjectures.

Describe how the graphs of the functions were affected (e.g., horizontal and vertical shifts, horizontal and vertical stretches, or reflections).

Use technology to explain possible effects on the graph from adding or multiplying the input or output of a function by a constant value.

Given the graph of a function and the graph of a translation, stretch, or reflection of that function, determine the value which was used to shift, stretch, or reflect the graph.

Teacher Vocabulary:

Composite functions

Horizontal and vertical shifts

Horizontal and vertical stretch

Reflections

Translations

Knowledge:

Students know:

Graphing techniques of functions.

Methods of using technology to graph functions

Skills:

Students are able to:

Accurately graph functions.

Check conjectures about how a parameter change in a function changes the graph and critique the reasoning of others about such shifts.

Identify shifts, stretches, or reflections between graphs.

Understanding:

Students understand that:

Graphs of functions may be shifted, stretched, or reflected by adding or multiplying the input or output of a function by a constant value.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: ALGI.23.1: Define dilation, rotation, reflection, translation, congruent and sequence.
ALGI.23.2: Identify rotations.
ALGI.23.3: Identify reflections.
ALGI.23.4: Identify translations.
ALGI.23.5: Use digital tools to formulate solutions to authentic problems (Ex: electronic graphing tools, probes, spreadsheets).

Prior Knowledge Skills:

Identify congruent figures.

Compare rotations to translations.

Compare reflections to rotations.

Compare translations to reflections.

Recognize translations (slides), rotations (turns), and reflections (flips).

Alabama Alternate Achievement Standards

AAS Standard: M.A.AAS.12.24 Given a simple linear function on a graph, select the model that represents an increase by equal amounts over equal intervals.

Mathematics MA2019 (2019) Grade: 9-12 Algebra I with Probability

30. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

b. Graph piecewise-defined functions, including step functions and absolute value functions.

c. Graph exponential functions, showing intercepts and end behavior.

Unpacked Content

Evidence Of Student Attainment:

Students:
Given a symbolic representation of a function (including linear, quadratic, absolute value, piecewise-defined functions, and exponential,

Produce an accurate graph (by hand in simple cases and by technology in more complicated cases) and justify that the graph is an alternate representation of the symbolic function.

Identify key features of the graph and connect these graphical features to the symbolic function, specifically for special functions:
quadratic or linear (intercepts, maxima, and minima) and piecewise-defined functions, including step functions and absolute value functions (descriptive features such as the values that are in the range of the function and those that are not).

Exponential (intercepts and end behavior).

Teacher Vocabulary:

x-intercept

y-intercept

Maximum

Minimum

End behavior

Linear function

Factorization

Quadratic function

Intercepts

Piece-wise function

Step function

Absolute value function

Exponential function

Domain

Range

Period

Midline

Amplitude

Zeros

Knowledge:

Students know:

Techniques for graphing.

Key features of graphs of functions.

Skills:

Students are able to:

Identify the type of function from the symbolic representation.

Manipulate expressions to reveal important features for identification in the function.

Accurately graph any relationship.

Understanding:

Students understand that:

Key features are different depending on the function.

Identifying key features of functions aid in graphing and interpreting the function.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: ALGI.30.1: Define piecewise-defined functions and step functions.
ALGI.30.2: Graph functions expressed symbolically by hand in simple cases.
ALGI.30.3: Graph functions expressed symbolically using technology for a more complicated case.

a.
ALGI.30.4: Graph quadratic functions showing maxima and minima.
ALGI.30.5: Graph quadratic functions showing intercepts.
ALGI.30.6: Graph linear functions showing intercepts.

b.
ALGI.30.7: Define square root, cube root, and absolute value function.
ALGI.30.8: Graph piecewise-defined functions.
ALGI.30.9: Graph step functions.
ALGI.30.10: Graph cube root functions.
ALGI.30.11: Graph square root functions.
ALGI.30.12: Graph absolute value functions.

c.
ALGI.30.13 Identify exponential numbers as repeated multiplication.
ALGI.30.14 Rewrite exponential numbers as repeated multiplication.

Prior Knowledge Skills:

Demonstrate how to plot points on a coordinate plane using ordered pairs from a table.

Graph a function given the slope-intercept form of an equation.

Recognize the absolute value of a rational number is its' distance from 0 on a vertical and horizontal number line.

Define absolute value and rational numbers.

Recall how to plot ordered pairs on a coordinate plane.

Name the pairs of integers and/or rational numbers of a point on a coordinate plane.

Alabama Alternate Achievement Standards

AAS Standard: M.A.AAS.12.30 Given the graph of a linear function, identify the intercepts, the maxima, and minima.