Government Standards

Common Core

Math - Grade 11

Assessment Exam - Common Core Math - High School: Functions
Interpreting Functions eTAP Lesson
Understand the concept of a function and use function notation.
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).
Function Notation
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
Direct Variation
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.
Linear Parametric equations
Interpret functions that arise in applications in terms of the context.
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.
Graphing Systems of Inequalities
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.
How to Use Graphs
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.
Graph linear and quadratic functions and show intercepts, maxima, and minima.
Simple Quadratic Functions
Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
Square Root Functions
Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
Graphs of Polynomial Functions
Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
Graphs of Exponential Functions
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.
Factoring Quadratics
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 Functions and Data
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.
Building Functions eTAP Lesson
Build a function that models a relationship between two quantities.
Determine an explicit expression, a recursive process, or steps for calculation from a context.
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.
Ratios and Proportions
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.
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.*
Arithmetic and Geometric Series
Build new functions from existing functions.
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.
Slope of a Line
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) =2 x3 or f(x) = (x+1)/(x-1) for x ≠ 1.
Inverses of Linear Functions
Verify by composition that one function is the inverse of another.
Inverses of Exponential Functions
Read values of an inverse function from a graph or a table, given that the function has an inverse.
Inverse Variation
Produce an invertible function from a non-invertible function by restricting the domain.
Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Direct and Inverse Variation
Linear, Quadratic, and Exponential Models eTAP Lesson
Construct and compare linear, quadratic, and exponential models and solve problems.
Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
Sums of Arithmetic Series
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).
Exponential and Logarithmic Equations
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
Using Logarithms to Model Data
Interpret expressions for functions in terms of the situation they model.
Interpret the parameters in a linear or exponential function in terms of a context.
Trigonometric Functions eTAP Lesson
Extend the domain of trigonometric functions using the unit circle.
Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
Measuring Angles

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.

Addition Formulas
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.
Sine and Cosine and the Unit Circle

Definitions - Sine and Cosine
Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Law of Sines

Law of Cosines
Model periodic phenomena with trigonometric functions.
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
Amplitude and Period
Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
Graphing Trigonometric Functions
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.
Tangent and Cotangent Functions
Prove and apply trigonometric identities.
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.
The Theorem of Pythagoras
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
Trigonometric Calculations

Finding Unknown Sides and Angles