# [F] Functions

Studentsâ€™ prior knowledge includes:

• Students define, evaluate and compare functions (grade 8)
• Students use functions to model relationships between quantities (grade 8)
• Students solve real-life and mathematical problems using numerical and algebraic expressions and equations (grade 7)
• Students represent and analyze quantitative relationships between dependent and independent variables (grade 6)
##### Interpreting Functions
• Understand the concept of a function and use function notation.
• Interpret functions that arise in applications in terms of the context.
• Analyze functions using different representations.
##### Building Functions
• Build a function that models a relationship between two quantities.
• Build new functions from existing functions.
##### Linear, Quadratic, and Exponential Models
• Construct and compare linear, quadratic, and exponential models and solve problems.
• Interpret expressions for functions in terms of the situation they model.
##### Trigonometric Functions
• Extend the domain of trigonometric functions using the unit circle.
• Model periodic phenomena with trigonometric functions.
• Prove and apply trigonometric identities.

## MAT-HS.F [F] Overview: Function

### MAT-HS.F-IF Domain: [F-IF] Interpreting Functions

• MAT-HS.F-IF.01 Understand that each element of the domain is exactly one element of the range
• MAT-HS.F-IF.02 Use, evaluate, and interpret function notation in terms of a context
• MAT-HS.F-IF.03 Recognize that sequences are functions whose domain is a subset of the integers
• MAT-HS.F-IF.04 Interpret graphs of functions that model a relationship between 2 quantities
• MAT-HS.F-IF.05 Relate the domain of a function to its graph and the relationship it describes
• MAT-HS.F-IF.06 Calculate and interpret the average rate of change of a function
• MAT-HS.F-IF.07 graph Functions expressed symbolically and show key features of the graph
• MAT-HS.F-IF.07.a Graph linear functions and show intercepts.
• F-IF.07.b Graph square root, cube root, and piecewisedefined functions
• F-IF.07.c Graph polynomial functions
• F-IF.07.d Graph rational functions
• MAT-HS.F-IF.07.e Graph exponential and logarithmic functions
• MAT-HS.F-IF.07.f graph a quadratic function and show intercepts, maxima and minima.
• MAT-HS.F-IF.08 Write a function defined by an expression in different but equivalent forms
• F-IF.08.a Use factoring and completing the square to show zeros/extreme values/symmetry
• F-IF.08.b Use properties of exponents to interpret expressions for exponential functions
• MAT-HS.F-IF.09 Compare properties of two Functions each represented in a different way

### MAT-HS.F-BF Domain: [F-BF] Building Functions

• MAT-HS.F-BF.01 Write a function that describes a relationship between two quantities
• F-BF.01.a Determine an expression/recursive process/steps for calculation from context
• F-BF.01.b Combine standard function types using arithmetic operations
• F-BF.01.c Compose functions
• MAT-HS.F-BF.02 Write arithmetic and geometric sequences, and translate between the two
• MAT-HS.F-BF.03 Identify effects on a graph, replacing f(x) with f(x) + k, f(xk), k(fx), f(x+k)
• MAT-HS.F-BF.04 Find inverse functions
• F-BF.04.a Solve f(x)
• F-BF.04.b Verify by composition that one function is the inverse of another
• F-BF.04.c Read values of an inverse function from a graph or a table
• F-BF.04.d Produce an invertible function from a noninvertible function
• MAT-HS.F-BF.05 Understand the inverse relationship between exponents and logarithms

### MAT-HS.F-LE Domain: [F-LE] Linear, Quadratic and Exponential Models*

• MAT-HS.F-LE.01 Know when to use linear vs exponential functions
• F-LE.01.a Prove linear functions grow by differences & exponential functions by factors
• F-LE.01.b Recognize where a quantity changes at a constant rate relative to another
• F-LE.01.c Recognize where a quantity grows/decays by constant percent relative to another
• MAT-HS.F-LE.02 Construct linear and exponential functions given a graph/relationship/pairs
• MAT-HS.F-LE.03 Observe that exponential increases are greater than linear increases
• MAT-HS.F-LE.04 For exponential models, express as a logarithm the solution to ab^ct = d
• MAT-HS.F-LE.05 Interpret parameters in a linear or exponential function in a context

### MAT-HS.F-TF Domain: [F-TF] Trigonometric Functions

• MAT-HS.F-TF.01 Understand radian measure of an angle
• MAT-HS.F-TF.02 Explain how the unit circle enables trigonometric functions to all real numbers
• MAT-HS.F-TF.03 Use special triangles to determine geometrically the values of sine, cosine
• MAT-HS.F-TF.04 Use the unit circle to explain symmetry (odd and even)
• MAT-HS.F-TF.05 Choose trigonometric functions to model periodic phenomena
• MAT-HS.F-TF.06 Understand that restricting a trigonometric function to a domain on which i
• MAT-HS.F-TF.07 Use inverse functions to solve trigonometric equations that arise in modeli
• MAT-HS.F-TF.08 Prove the Pythagorean identity sin^2(?) + cos^2(?) = 1
• MAT-HS.F-TF.09 Prove the addition and subtraction formulas for sine, cosine, and tangent a

### A Sample of HS Math Concept Functions to Become Ready for College and Career

#### Interpreting Functions

• Understand the concept of a function and use function notation.
• Interpret functions that arise in applications in terms of the context.
• Analyze functions using different representations.

#### Building Functions

• Build a function that models a relationship between two quantities.
• Build new functions from existing functions.

#### Linear, Quadratic, and Exponential Models

• Construct and compare linear, quadratic, and exponential models and solve problems.
• Interpret expressions for functions in terms of the situation they model.

#### Trigonometric Functions

• Extend the domain of trigonometric functions using the unit circle.
• Model periodic phenomena with trigonometric functions.
• Prove and apply trigonometric identities.