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)

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.

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.

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.