1: Interpreting Functions

1.1: Understand the concept of a function and use function notation.

KY.HS.F.1: Understand properties and key features of functions and the different ways functions can be represented.

KY.HS.F.1.a: 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.

Exponential Functions
Introduction to Functions
Linear Functions
Logarithmic Functions
Points, Lines, and Equations
Radical Functions

KY.HS.F.1.b: Using appropriate 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
Introduction to Exponential Functions
Points, Lines, and Equations
Quadratics in Polynomial Form
Radical Functions

KY.HS.F.1.c: 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.

Absolute Value with Linear Functions
Cosine Function
Distance-Time Graphs
Distance-Time and Velocity-Time Graphs
Exponential Functions
Function Machines 3 (Functions and Problem Solving)
General Form of a Rational Function
Graphs of Polynomial Functions
Introduction to Exponential Functions
Logarithmic Functions
Points, Lines, and Equations
Quadratics in Factored Form
Quadratics in Polynomial Form
Quadratics in Vertex Form
Radical Functions
Rational Functions
Sine Function
Tangent Function

KY.HS.F.1.d: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

General Form of a Rational Function
Introduction to Functions
Logarithmic Functions
Logarithmic Functions: Translating and Scaling
Radical Functions
Rational Functions

KY.HS.F.1.e: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

General Form of a Rational Function
Graphs of Polynomial Functions
Linear Functions
Logarithmic Functions
Quadratics in Polynomial Form
Quadratics in Vertex Form

KY.HS.F.2: Recognize that arithmetic and geometric sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

Arithmetic Sequences
Arithmetic and Geometric Sequences
Geometric Sequences

1.2: Interpret functions that arise in applications in terms of the context.

KY.HS.F.3: Understand average rate of change of a function over an interval.

KY.HS.F.3.a: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval.

Cat and Mouse (Modeling with Linear Systems)
Distance-Time Graphs
Distance-Time and Velocity-Time Graphs
Slope

KY.HS.F.3.b: Estimate the rate of change from a graph.

Cat and Mouse (Modeling with Linear Systems)
Slope

1.3: Analyze functions using different representations.

KY.HS.F.4: Graph functions expressed symbolically and show key features of the graph, with and without using technology (computer, graphing calculator).

KY.HS.F.4.a: Graph linear and quadratic functions and show intercepts, maxima and minima.

Absolute Value with Linear Functions
Cat and Mouse (Modeling with Linear Systems)
Exponential Functions
Graphs of Polynomial Functions
Linear Functions
Point-Slope Form of a Line
Points, Lines, and Equations
Polynomials and Linear Factors
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
Zap It! Game

KY.HS.F.4.b: Graph square root, cube root and absolute value functions.

Absolute Value Equations and Inequalities
Absolute Value with Linear Functions
Radical Functions
Translating and Scaling Functions

KY.HS.F.4.c: Graph polynomial functions, identifying zeros when suitable factorizations are available and showing end behavior.

Graphs of Polynomial Functions
Polynomials and Linear Factors
Quadratics in Factored Form
Roots of a Quadratic
Zap It! Game

KY.HS.F.4.d: Graph exponential and logarithmic functions, showing intercepts and end behavior.

Cosine Function
Exponential Functions
Introduction to Exponential Functions
Logarithmic Functions
Logarithmic Functions: Translating and Scaling
Sine Function
Tangent Function
Translating and Scaling Sine and Cosine Functions

KY.HS.F.4.e: Graph trigonometric functions, showing period, midline and amplitude.

Cosine Function
Exponential Functions
Introduction to Exponential Functions
Logarithmic Functions
Logarithmic Functions: Translating and Scaling
Sine Function
Tangent Function
Translating and Scaling Sine and Cosine Functions

KY.HS.F.4.f: Graph piecewise functions, including step functions.

Absolute Value with Linear Functions
Radical Functions
Translating and Scaling Functions

KY.HS.F.4.g: Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available and showing end behavior.

General Form of a Rational Function
Rational Functions

KY.HS.F.5: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

KY.HS.F.5.a: Identify zeros, extreme values and symmetry of the graph within the context of a quadratic function.

Modeling the Factorization of x²+bx+c
Quadratics in Factored Form
Quadratics in Polynomial Form
Quadratics in Vertex Form
Roots of a Quadratic
Zap It! Game

KY.HS.F.5.b: Use the properties of exponents to interpret expressions for exponential functions and classify the exponential function as representing growth or decay.

Compound Interest
Exponential Functions
Exponential Growth and Decay
Introduction to Exponential Functions

2: Building Functions

2.1: Build a function that models a relationship between two quantities.

KY.HS.F.6: Write a function that describes a relationship between two quantities.

KY.HS.F.6.a: Determine an explicit expression, a recursive process, or steps for calculation from a context.

Arithmetic Sequences
Arithmetic and Geometric Sequences
Geometric Sequences

KY.HS.F.6.b: Combine standard function types using arithmetic operations.

Addition and Subtraction of Functions

KY.HS.F.7: Use arithmetic and geometric sequences to model situations and scenarios.

Arithmetic Sequences
Arithmetic and Geometric Sequences
Geometric Sequences

KY.HS.F.7.a: Use formulas (explicit and recursive) to generate terms for arithmetic and geometric sequences.

Arithmetic Sequences
Arithmetic and Geometric Sequences
Geometric Sequences

KY.HS.F.7.b: Write formulas to model arithmetic and geometric sequences and apply those formulas in realistic situations.

Arithmetic Sequences
Arithmetic and Geometric Sequences
Geometric Sequences

KY.HS.F.7.c: Translate between recursive and explicit formulas.

Arithmetic Sequences
Geometric Sequences

2.2: Build new functions from existing functions.

KY.HS.F.8: Understand the effects of transformations on the graph of a function.

KY.HS.F.8.a: 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.

Absolute Value with Linear Functions
Exponential Functions
Introduction to Exponential Functions
Logarithmic Functions
Logarithmic Functions: Translating and Scaling
Quadratics in Polynomial Form
Quadratics in Vertex Form
Radical Functions
Rational Functions
Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions
Translations
Zap It! Game

KY.HS.F.8.b: Experiment with cases and illustrate an explanation of the effects on the graph using technology.

Absolute Value with Linear Functions
Exponential Functions
Introduction to Exponential Functions
Logarithmic Functions
Logarithmic Functions: Translating and Scaling
Quadratics in Polynomial Form
Quadratics in Vertex Form
Radical Functions
Rational Functions
Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions
Translations
Zap It! Game

KY.HS.F.9: Find inverse functions.

KY.HS.F.9.a: Given the equation of an invertible function, find the inverse.

Logarithmic Functions

KY.HS.F.9.c: Read values of an inverse function from a graph or a table, given that the function has an inverse.

Logarithmic Functions

KY.HS.F.9.d: Produce an invertible function from a non-invertible function by restricting the domain.

Logarithmic Functions

KY.HS.F.10: Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents with the use of technology.

Logarithmic Functions

3: Linear, Quadratic and Exponential Functions

3.1: Construct and compare linear, quadratic and exponential models and solve problems.

KY.HS.F.11: Distinguish between situations that can be modeled with linear functions and with exponential functions.

KY.HS.F.11.a: Recognize and justify 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
Compound Interest
Direct and Inverse Variation
Exponential Functions
Introduction to Exponential Functions
Slope-Intercept Form of a Line

KY.HS.F.11.b: Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

Arithmetic Sequences
Arithmetic and Geometric Sequences
Compound Interest
Direct and Inverse Variation
Distance-Time Graphs
Distance-Time and Velocity-Time Graphs
Linear Functions
Slope-Intercept Form of a Line

KY.HS.F.11.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

KY.HS.F.12: 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).

Absolute Value with Linear Functions
Arithmetic Sequences
Arithmetic and Geometric Sequences
Compound Interest
Exponential Functions
Exponential Growth and Decay
Function Machines 1 (Functions and Tables)
Function Machines 2 (Functions, Tables, and Graphs)
Function Machines 3 (Functions and Problem Solving)
Geometric Sequences
Introduction to Exponential Functions
Linear Functions
Logarithmic Functions
Point-Slope Form of a Line
Points, Lines, and Equations
Slope-Intercept Form of a Line
Standard Form of a Line

KY.HS.F.13: 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
Compound Interest
Introduction to Exponential Functions

3.2: Interpret expressions for functions in terms of the situation they model.

KY.HS.F.14: Interpret the parameters in a linear or exponential function in terms of a context.

Arithmetic Sequences
Compound Interest
Distance-Time Graphs
Distance-Time and Velocity-Time Graphs
Exponential Growth and Decay
Introduction to Exponential Functions

4: Trigonometric Functions

4.1: Extend the domain of trigonometric functions using the unit circle.

KY.HS.F.15: Understand the relationship of radian measure of an angle to its arc length.

Cosine Function
Sine Function
Tangent Function

KY.HS.F.16: Understand and use the unit circle.

KY.HS.F.16.a: 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

KY.HS.F.16.b: 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
Sum and Difference Identities for Sine and Cosine
Tangent Function
Translating and Scaling Sine and Cosine Functions

KY.HS.F.16.c: Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Cosine Function
Sine Function
Tangent Function
Translating and Scaling Sine and Cosine Functions

4.2: Model periodic phenomena with trigonometric functions.

KY.HS.F.17: Choose trigonometric functions to model periodic phenomena with specified period, midline and amplitude.

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

4.3: Prove and apply trigonometric identities.

KY.HS.F.20: Proving identities and formulas within the context of trigonometry.

KY.HS.F.20.a: Prove the Pythagorean identity and use it to find sin(theta), cos(theta), or tan(theta) given sin(theta), cos(theta), or tan(theta) and the quadrant of the angle.

Cosine Function
Simplifying Trigonometric Expressions
Sine Function
Sine, Cosine, and Tangent Ratios
Tangent Function

KY.HS.F.20.b: 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