F-IF: Interpreting Functions

F-IF.A: Understand the concept of a function and use function notation.

F-IF.A.1: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Introduction to Functions
Linear Functions
Points, Lines, and Equations

F-IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Absolute Value with Linear Functions
Exponential Functions
Points, Lines, and Equations
Quadratics in Polynomial Form

F-IF.A.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

Arithmetic Sequences
Geometric Sequences

F-IF.B: Interpret functions that arise in applications in terms of the context (linear, quadratic, exponential, rational, polynomial, square root, cube root, trigonometric, logarithmic).

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.

Absolute Value with Linear Functions
Exponential Functions
General Form of a Rational Function
Graphs of Polynomial Functions
Logarithmic Functions
Quadratics in Factored Form
Quadratics in Polynomial Form
Quadratics in Vertex Form
Radical Functions
Slope-Intercept Form of a Line
Standard Form of a Line

F-IF.B.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Absolute Value with Linear Functions
Exponential Growth and Decay
Points, Lines, and Equations

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.

Distance-Time Graphs
Distance-Time Graphs - Metric
Distance-Time and Velocity-Time Graphs
Distance-Time and Velocity-Time Graphs - Metric
Slope

F-IF.C: 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.

F-IF.C.7.a: Graph linear and quadratic functions and show intercepts, maxima, and minima.

Point-Slope Form of a Line
Quadratics in Factored Form
Quadratics in Polynomial Form
Quadratics in Vertex Form
Roots of a Quadratic
Slope-Intercept Form of a Line
Standard Form of a Line
Translating and Scaling Functions

F-IF.C.7.b: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Absolute Value with Linear Functions
Radical Functions
Translating and Scaling Functions

F-IF.C.7.c: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

Graphs of Polynomial Functions
Polynomials and Linear Factors

F-IF.C.7.d: Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

General Form of a Rational Function
Rational Functions

F-IF.C.7.e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Exponential Functions
Introduction to Exponential Functions
Logarithmic Functions
Logarithmic Functions: Translating and Scaling
Translating and Scaling Functions

F-IF.C.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

F-IF.C.8.a: Use the process of factoring and/or completing the square in quadratic and polynomial functions, where appropriate, to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Quadratics in Factored Form

F-IF.C.8.b: Use the properties of exponents to interpret expressions for exponential functions. Apply to financial situations such as identifying appreciation and depreciation rate for the value of a house or car some time after its initial purchase. V sub n = P(1 + r)^n

Compound Interest
Exponential Growth and Decay

F-IF.C.9: Translate among different representations of functions (algebraically, graphically, numerically in tables, or by verbal descriptions). Compare properties of two functions each represented in a different way.

Exponential Functions
General Form of a Rational Function
Graphs of Polynomial Functions
Logarithmic Functions
Quadratics in Factored Form
Quadratics in Polynomial Form
Radical Functions

F-IF.C.10: Given algebraic, numeric and/or graphical representations of functions, recognize the function as polynomial, rational, logarithmic, exponential, or trigonometric.

Graphs of Polynomial Functions
Logarithmic Functions

F-BF: Building Functions

F-BF.A: Build a function that models a relationship between two quantities.

F-BF.A.1: Write a function (linear, quadratic, exponential, simple rational, radical, logarithmic, and trigonometric) that describes a relationship between two quantities.

F-BF.A.1.a: Determine an explicit expression, a recursive process, or steps for calculation from a context.

Arithmetic Sequences
Arithmetic and Geometric Sequences
Geometric Sequences

F-BF.A.1.b: Combine standard function types using arithmetic operations.

Addition and Subtraction of Functions

F-BF.A.2: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Arithmetic Sequences
Arithmetic and Geometric Sequences
Geometric Sequences

F-BF.B: Build new functions from existing functions.

F-BF.B.3: Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. (Include linear, quadratic, exponential, absolute value, simple rational and radical, logarithmic and trigonometric functions.) Utilize technology to experiment with cases and illustrate an explanation of the effects on the graph. (Include recognizing even and odd functions from their graphs and algebraic expressions for them.)

Exponential Functions
Graphs of Polynomial Functions
Introduction to Exponential Functions
Logarithmic Functions: Translating and Scaling
Quadratics in Polynomial Form
Quadratics in Vertex Form
Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions
Zap It! Game

F-BF.B.4: Find inverse functions algebraically and graphically.

F-BF.B.4.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. (Include linear and simple polynomial, rational, and exponential functions.)

Logarithmic Functions
Radical Functions

F-BF.B.4.c: Read values of an inverse function from a graph or a table, given that the function has an inverse.

Logarithmic Functions

F-BF.B.5: Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Logarithmic Functions

F-LE: Linear, Quadratic, and Exponential Models

F-LE.A: Construct and compare linear, quadratic, and exponential models and solve problems.

F-LE.A.1: Distinguish between situations that can be modeled with linear functions and with exponential functions.

F-LE.A.1.a: Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

Arithmetic and Geometric Sequences

F-LE.A.1.b: Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

Arithmetic Sequences
Arithmetic and Geometric Sequences

F-LE.A.1.c: Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Arithmetic and Geometric Sequences
Compound Interest
Exponential Growth and Decay
Geometric Sequences

F-LE.A.2: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (including reading these from a table).

Arithmetic Sequences
Arithmetic and Geometric Sequences
Compound Interest
Exponential Functions
Exponential Growth and Decay
Geometric Sequences
Introduction to Exponential Functions
Point-Slope Form of a Line
Slope-Intercept Form of a Line
Standard Form of a Line

F-LE.A.3: Observe, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Arithmetic and Geometric Sequences

F-LE.A.4: 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.

Logarithmic Functions

F-LE.B: Interpret expressions for functions in terms of the situation they model.

F-LE.B.5: Interpret the parameters in a linear or exponential function (of the form f(x) = b^x + k) in terms of a context.

Compound Interest
Exponential Growth and Decay
Slope-Intercept Form of a Line
Standard Form of a Line

F-TF: Trigonometric Functions

F-TF.A: Extend the domain of trigonometric functions using the unit circle.

F-TF.A.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

Cosine Function
Radians
Sine Function
Tangent Function

F-TF.A.2: Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

Cosine Function
Sine Function
Tangent Function

F-TF.A.3: Use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, and tangent for pi – x, pi + x, and 2pi – x in terms of their values for x, where x is any real number.

Cosine Function
Sine Function
Tangent Function

F-TF.A.4: Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Cosine Function
Sine Function
Tangent Function

F-TF.B: Model periodic phenomena with trigonometric functions.

F-TF.B.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

Sine Function
Sound Beats and Sine Waves
Translating and Scaling Sine and Cosine Functions
Waves

F-TF.C: Prove and apply trigonometric identities.

F-TF.C.8: Prove the Pythagorean identity sin²(theta) + cos²(theta) = 1 and use it to find sin(theta), cos(theta), or tan(theta) given sin(theta), cos(theta), or tan(theta) and the quadrant.

Cosine Function
Sine Function

F-TF.C.9: Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Sum and Difference Identities for Sine and Cosine