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Common Core Mapping for High School: Functions




 

Functions Overview

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 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

  • Related Topics:
    Common Core for Mathematics

    Common Core Mapping for High School: Functions

    Interpreting Functions

    Code

    Standard

    Lessons

    Worksheets

    Games

    HSF-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).

    Functions (Domain & Range)

    Find Domain & Range of Functions

    Domain of a function

    Range of a function

    Recognizing functions 2

     

    HSF-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.

    Function Notation, Applications and Operations

    Understanding function notation

    Evaluating expressions with function notation

     

    HSF-IF.A.3

    Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

    Functions and Sequences

    Recursive and explicit functions

     

    HSF-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 Features of Functions

    Modeling Functions

    Interpreting features of functions

    Recognizing features of functions

    Positive and negative parts of functions

     

    HSF-IF.B.5

    Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For 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.

    Interpret Domain of a Function

    Domain and range from graph

    Domain of a function

    Range of a function

     

    HSF-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.

    Average Rate of Change

    Average rate of change

     

    HSF-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.

    See Below

    Graphing linear equations

    Line graph intuition

    Linear function intercepts

    HSF-IF.C.7a

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

    Graph Linear and Quadratic Functions

    Graphing parabolas in standard form

    Graphing parabolas in vertex form

    Graphing parabolas in all forms

     

    HSF-IF.C.7b

    Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

    Graph Square Root & Cube Root Functions

    Graph Piecewise-Defined Functions

    Graph Absolute Value Functions

    Graphs of radical functions

    Graphs of absolute value functions

    Graphs of piecewise functions

     

    HSF-IF.C.7c

    Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

    Graph Polynomial Functions

    Polynomial Graphs

    Graph Polynomial Functions

    HSF-IF.C.7d

    (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

    Graph Rational Functions

    Rational Function Graphs

    Graph Rational Functions

    HSF-IF.C.7e

    Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

    Graph Exponential & Logarithmic Functions

    Graph Trigonometric Functions

    Features of trigonometric functions

    Graphs of trigonometric functions

    Graphs of exponentials and logarithms

     

    HSF-IF.C.8

    Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

    See Below

    Rewriting quadratic expressions

    HSF-IF.C.8a

    Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

    Quadratic Function Applications

    Solve Quadratics by Factoring

    Solve Quadratics by Completing the Square

    HSF-IF.C.8b

    Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.

    Exponential Function Applications

    Exponential Functions

    HSF-IF.C.9

    Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

    Compare Properties of Functions

    Comparing features of functions

     



    Building Functions

    Code

    Standard

    Lessons

    Worksheets

    Games

    HSF-BF.A.1

    Write a function that describes a relationship between two quantities.

    See Below


     

    HSF-BF.A.1a

    Determine an explicit expression, a recursive process, or steps for calculation from a context.

    Explicit & Recursive Word Problems

    Recursive and explicit functions

     

    HSF-BF.A.1b

    Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

    Modeling with equations and inequalities

    Modeling with one-variable equations and inequalities

    HSF-BF.A.1c

    (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

    Composite Functions

    Evaluate Composite Functions

    Modeling with composite functions

    HSF-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.

    Recursive and Explicit Functions

    Recursive and explicit functions

     

    HSF-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.

    Function Transformation

    Even and Odd Functions

    Shifting and reflecting functions

    Even and odd functions

     

    HSF-BF.B.4

    Find inverse functions.

    See Below


     

    HSF-BF.B.4a

    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. For example, f(x) =2x3 or f(x) = (x+1)/(x?1) for x ≠ 1.

    Inverse Functions

    Inverses of linear functions

    Understand inverses of functions

     

    HSF-BF.B.4b

    (+) Verify by composition that one function is the inverse of another.

    Verify Inverse Functions

    Understand inverses of functions

    HSF-BF.B.4c

    (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.

    Inverse Functions (Graph, Table)

    Understand inverses of functions

    HSF-BF.B.4d

    (+) Produce an invertible function from a non-invertible function by restricting the domain.

    Inverse Function (Restricted Domain)

    Understand inverses of functions

    HSF-BF.B.5

    (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

    Exponents and Logarithms

    Evaluating logarithms

    Evaluating logarithms 2

    Undestanding logarithms as inverse exponentials

    Operations with logarithms

     


     

    Linear, Quadratic, and Exponential Models

    Code

    Standard

    Lessons

    Worksheets

    Games

    HSF-LE.A.1

    Distinguish between situations that can be modeled with linear functions and with exponential functions.

    Linear & Exponential Word Problems

    Exponential Word Problems

    HSF-LE.A.1a

    Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

    Compare Linear & Exponential Functions

    Understand linear and exponential models

    HSF-LE.A.1b

    Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

    Understand linear and exponential models

    HSF-LE.A.1c

    Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

    Growth & Decay Functions

    Understand linear and exponential models

    HSF-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 (include reading these from a table).

    Construct Linear and Exponential Functions

    Construct linear & exponential functions

     

    HSF-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.

    Increase Exponentially

    Compare growth rates of exponentials and polynomials

    HSF-LE.A.4

    For exponential models, express as a logarithm the solution to abc = d where ac, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

    Solve Exponential Equations

    Use logarithms to solve exponential equations

    HSF-LE.B.5

    Interpret the parameters in a linear or exponential function in terms of a context.

    Parameters in Linear or Exponential Functions

    Modeling with exponential functions

     


    Trigonometric Functions

    Code

    Standard

    Lessons

    Worksheets

    Games

    HSF-TF.A.1

    Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

    Radian Measure

    Radians and arc length

    Convert between Radians & Degrees

     

    HSF-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.

    Unit Circle

    Unit circle

    Unit circle trigonometry

    HSF-TF.A.3

    (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for x, π + x, and 2π ? x in terms of their values for x, where x is any real number.

    Special Triangles & Unit Circle

    Trigonometric functions of special angles

    Unit Circle

    HSF-TF.A.4

    (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

    Unit Circle (Symmetry, Periodicity)

    Symmetry and periodicity of trigonometric functions

    HSF-TF.B.5

    Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

    Modeling with Trigonometric Functions

    Modeling with periodic functions

    Modeling with periodic functions 2

    HSF-TF.B.6

    (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

    Inverse Function

    Understand inverse trigonometric functions

    Inverse Function


    HSF-TF.B.7

    (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

    Use Inverse Trig Functions

    Inverse trig word problems

    Inverse Function

    HSF-TF.C.8

    Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

    Pythagorean Identity

    Circles and Pythagorean identities

    Pythagorean identities

     

    HSF-TF.C.9

    (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

    Addition & Subtraction Formulas

    Addition and subtraction trig identities

    Understand angle addition formulas

    Applying angle addition formulas

     

    Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.


    You can use the free Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.


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