Algebra 2 (& GT)

Unit 5: Exponential and Logarithmic Functions

What will my child learn?

Students will:

Understand the concept of a function and use function notation.
F.IF.A.3
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers (SAT® Content - PAM.14).

Build a function that models a relationship between two quantities.
F.BF.A.2
Write geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
Construct and compare linear and exponential models and solve problems.
F.LE.A.2
Construct exponential functions, including geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table) (SAT® Content - PAM.01). 
Interpret expressions for functions in terms of the situations they model.
F.LE.B.5
Interpret the parameters in an exponential function in terms of a context (SAT® Content - PAM.12). 
Write expressions in equivalent forms to solve problems.
A.SSE.B.4
Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. 
Build new functions from existing functions.
F.BF.B.4
Find inverse functions. Focus on linear functions, but consider simple situations where the domain of the function must be restricted in order for the inverse to exist, such as f(x) = x^2, x > 0.
a. Solve an equation of the form f(x) = c. For example, f(x) = 2x^3 or f(x) = (x + 1)/(x – 1) for x ≠ 1.
Analyze functions using different representations.
F.IF.C.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*  
e. Graph exponential and logarithmic functions, showing intercepts and end behavior.
Interpret functions that arise in applications in terms of a context.
F.IF.B.4
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.* 
Interpret linear and exponential functions that arise in applications in terms of a context.
F.IF.B.6
Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval.  Estimate the rate of change from a graph.  
Analyze functions using different representations.
F.IF.C.9
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 
Build a function that models a relationship between two quantities.
F.BF.A.1
Write a function that describes a relationship between two quantities.* 
b.  Combine standard function types using arithmetic operations. 
Write expressions in equivalent forms to solve problems
A.SSE.B.3
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
c.  Use the properties of exponents to transform expressions for exponential functions.
Build new functions from existing functions.
F.BF.B.3
Identify the effect on the graph of 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 of 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 expressions for them. 
Represent and solve equations graphically.
A.REI.D.11
Explain why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations.  Include cases where f(x) and/or g(x) are linear, exponential, and logarithmic functions.* 
Construct and compare linear, quadratic, and exponential models and solve problems.
F.LE.A.4
For exponential models, express as a logarithm the solution to a b^ct = d, where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
What are some signs of student mastery?
  • Builds functions that model mathematical and contextual situations, including those requiring multiple trigonometric functions, sequences and combinations of these and other functions, and uses the models to solve, interpret and generalize about problems.
  • Constructs linear and exponential function models in multi-step contextual problems.
  • Uses mathematical properties and relationships to reveal key features of polynomial, rational, trigonometric and logarithmic functions to sketch graphs and identify characteristics of the relationship between two quantities.
  • Identifies how changing the parameters of the function impacts key features of the graph.
  • Graphs polynomial, trigonometric and logarithmic functions, showing key features.
  • Builds functions that model mathematical and contextual situations, including those requiring multiple trigonometric functions, sequences and combinations of these and other functions, and uses the models to solve, interpret and generalize about problems.
  • Determines equivalent forms of quadratic and exponential (with integer domain) expressions and functions to reveal and explain their properties.
  • Given a scenario, determines the most appropriate form of a quadratic or exponential (with integer domain) function.
  • Calculates and interprets the average rate of change of polynomial, exponential, logarithmic or trigonometric functions (presented symbolically or as a table) over a specified interval, and estimates the rate of change from a graph.
  • Compares rates of change associated with different intervals.
  • Identifies the effects of multiple transformations on graphs of linear and quadratic functions and finds the value of k given a transformed graph.
  • Experiments with cases using technology.
  • Given the equation of a transformed linear or quadratic function, creates an appropriate graph.
  • Compares the properties of two functions represented in multiple ways, limited to linear, exponential (with domains in the integers), quadratic, square root, absolute value, cube root, piece-wise and step.
  • Finds the solutions to two polynomial functions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations.
  • Solves non-routine mathematical equations directly and indirectly, using structure, technology, graphs, formulas, tables of values and successive approximations, and gives examples of how extraneous solutions may arise.
Tools & Technology
Desmos is a free online graphing calculator that works on any computer or tablet without requiring any downloads. A FREE Desmos iPad app is available too!

Try the Geometric Sequence Calculator!  

Interested in exploring more with series?  Check out the Convergence Calculator.  

If you'd like to visualize functions and their inverses, look at the Inverse Function Calculator from Wolfram!