- Course type: Self-paced
- Available Lessons: 34
- Average Lesson Length: 8 min
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Watch a preview:chapter 1 / lesson 1Functions: Identification, Notation & Practice Problems
Course SummaryThe lessons included in this Common Core Math: Functions course give teachers additional materials to help students grasp important functions concepts with ease. Let entertaining lessons, short quizzes and practice exams supplement your curriculum and help you meet the Common Core State Standards.
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Course Practice TestCheck your knowledge of this course with a 50-question practice test.
- Comprehensive test covering all topics
- Detailed video explanations for wrong answers
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About the Course
To ensure all students at each grade level receive the same education and learn the same basic concepts, parents, teachers and other representatives of communities developed the Common Core State Standards. These standards manage the topics and concepts taught by grade as well as ensure students are evaluated based on their understanding of the concepts. We provide chapters and lessons within this course on mathematical functions that correlate to the standards for this concept. The video lessons are short, and each lesson breaks down the topic into simple segments so that students have a better chance of understanding and retaining the information. The following principles related to functions are covered in this course:
- Evaluating and composing functions
- Graphing inverse functions
- Classifying arithmetic sequences
- Identifying parts of a graph
- Working with linear equations
- Solving problems with quadratics
- Shifting and reflecting transformations
- Graphing and solving logarithmic functions
- Introducing trigonometry concepts
In addition to providing video lessons that can be used as supplementary classroom materials or tutorials for homework assignments, the lessons each contain a correlating quiz that can be used to ensure students fully understand the principles and objectives being discussed. Teachers can engage students more fully by employing these videos within their teaching methods. Each chapter also includes helpful tips that you can use to further interest students in the topics being covered.
Domains and Ranges (CCSS.Math.Content.HSF-IF.A.1)
Standard: 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).
Lessons cover identifying functions and notation, and explain domains and ranges in a function.
Evaluating Functions (CCSS.Math.Content.HSF-IF.A.2)
Standard: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
Use these lessons to teach students how to evaluate functions and discuss application problems using functions.
Sequences as Functions (CCSS.Math.Content.HSF-IF.A.3)
Standard: 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 greater than or equal to 1.
These lessons identify a mathematical sequence, define sequences as mathematical functions and introduce students to recursive sequences.
Intro to Graphing (CCSS.Math.Content.HSF-IF.B.4)
Standard: 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.
Lessons for this standard describe the different parts of a graph, explain graphs as relationships between two quantities, and show students how to sketch graphs from tables or verbal descriptions of relationships.
Graphing Functions (CCSS.Math.Content.HSF-IF.B.5)
Standard: 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.
This collection includes a lesson on relating domains to graphs of functions.
Rate of Change (CCSS.Math.Content.HSF-IF.B.6)
Standard: 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.
These lessons show how to find the average rate of change from a graph or table and how to calculate the average rate of change from an equation.
Graphing, Parabolas and Quadratic Functions (CCSS.Math.Content.HSF-IF.C.7a)
Standard: Graph linear and quadratic functions and show intercepts, maxima, and minima.
Lessons for this standard discuss graphs of equations, linear equations and abstract algebra. Students can learn how to graph basic functions, undefined slopes and zero slope. Lessons also introduce parabolas and quadratic functions.
Graphing Functions (CCSS.Math.Content.HSF-IF.C.7b)
Standard: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
Video lessons teach how to graph square root and cube root functions. Students are introduced to piecewise-defined functions, step functions and absolute value functions, and taught how to graph them.
Standard: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
Lessons explain the properties and exponential of a polynomial as well as how to factor and simplify polynomials. They also teach how to graph monomials and polynomials.
Rational Functions (CCSS.Math.Content.HSF-IF.C.7d)
Standard: Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
These lessons define rational functions and show students how to find vertical and horizontal asymptotes of rational functions. They also explain calculating for zero, finding the end behavior and y-intercepts and graphing rational functions.
Exponential Functions, Logarithms, Sine & Cosine (CCSS.Math.Content.HSF-IF.C.7e)
Standard: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
Videos teach how to graph exponential and logarithmic functions, and discuss the transformations of the graphs. Students are introduced to sine, cosine and tangents and shown how to graph them.
Quadratic Functions (CCSS.Math.Content.HSF-IF.C.8a)
Standard: 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.
Use these videos to discuss the zeros of quadratic functions and how to solve problems using quadratic functions.
Standard: 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.
These lessons go over the five main exponent properties, defining zero and negative exponents, the explanation of an exponential function, and exponential growth vs. decay.
Algebraic & Graphical Functions (CCSS.Math.Content.HSF-IF.C.9)
Standard: 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.
This lesson shows students how to compare functions represented graphically and algebraically.
Explicit Expressions, Recursive Process & Context (CCSS.Math.Content.HSF-BF.A.1a)
Standard: Determine an explicit expression, a recursive process, or steps for calculation from a context.
These lessons compare explicit expressions and recursive processes. Students also learn how to calculate a function from a context.
Calculating Functions (CCSS.Math.Content.HSF-BF.A.1b)
Standard: 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.
Take advantage of these lessons to teach students how to add, subtract, multiply and divide functions. Use the practice problems to show students how to apply function operations.
Composing Functions (CCSS.Math.Content.HSF-BF.A.1c)
Standard: 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.
This lesson explains how to compose functions.
Arithmetic & Geometric Sequences (CCSS.Math.Content.HSF-BF.A.2)
Standard: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
These lessons describe how to find and classify arithmetic and geometric sequences, and use these sequences to model situations.
Function Transformations (CCSS.Math.Content.HSF-BF.B.3)
Standard: 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.
Lessons for this standard explain shifting, reflecting, stretching and compressing function transformations, as well as how to put it all together.
Inverse Functions (CCSS.Math.Content.HSF-BF.B.4a and CCSS.Math.Content.HSF-BF.B.4b)
Standards: 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 x to the 3rd power or f(x) = (x+1)/(x-1) for x is not equal to 1. Verify by composition that one function is the inverse of another.
These lessons introduce inverse functions, explain graphing inverse functions, and include practice problems for finding and verifying inverse functions.
Inverse Functions in a Table or Graph (CCSS.Math.Content.HSF-BF.B.4c)
Standard: Read values of an inverse function from a graph or a table, given that the function has an inverse.
This lesson describes how to find inverses of functions represented by tables or graphs.
Invertible Function (CCSS.Math.Content.HSF-BF.B.4d)
Standard: Produce an invertible function from a non-invertible function by restricting the domain.
This lesson shows students how to produce and invertible function from a non-invertible function by restricting the domain.
Standard: Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
These lessons discuss logarithmic and exponential functions as inverses of each other. They also identify logarithmic properties, explain how to solve logarithmic equations, and include practice problems.
Growth Patterns (CCSS.Math.Content.HSF-LE.A.1a)
Standard: Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
This lesson shows students how to identify linear vs. exponential growth patterns.
Linear Growth (CCSS.Math.Content.HSF-LE.A.1b)
Standard: Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
Use these lessons to explain lines as models for growth and real-world linear growth.
Exponential Growth (CCSS.Math.Content.HSF-LE.A.1c)
Standard: Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
This lesson helps explain real-world exponential growth to students.
Constructing Functions (CCSS.Math.Content.HSF-LE.A.2)
Standard: 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).
Lessons that cover this standard show students how to write rules to describe arithmetic and geometric sequences; find the equation of a line from graphs, points or descriptions; and find and exponential equation from graphs, points or descriptions.
Exponential Increase (CCSS.Math.Content.HSF-LE.A.3)
Standard: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
This lesson shows how exponential growth has the ability to become enormous.
Exponential Equations & Logarithms (CCSS.Math.Content.HSF-LE.A.4)
Standard: For exponential models, express as a logarithm the solution to ab to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
Use this lesson to show students how to solve exponential equations using logarithms.
Standard: Interpret the parameters in a linear or exponential function in terms of a context.
This lesson explains how to interpret the coefficients of an exponential equation in context.
Converting Radians & Degrees (CCSS.Math.Content.HSF-TF.A.1)
Standard: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
Students can learn how to convert between radians and degrees with this lesson.
Unit Circle (CCSS.Math.Content.HSF-TF.A.2)
Standard: 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.
The lesson on the unit circle shows how to extend trigonometric functions beyond the first quadrant.
Special Triangles (CCSS.Math.Content.HSF-TF.A.3)
Standard: 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 x, pi + x, and 2pi - x in terms of their values for x, where x is any real number.
Show students how to use special triangles and the unit circle to find values for sine, cosine and tangent and complete practice problems.
Odd, Even & Periodicity of Trigonometric Functions (CCSS.Math.Content.HSF-TF.A.4)
Standard: Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
These lessons discuss odd and even trigonometric functions, as well as the periodicity of trigonometric functions.
Graphing Trigonometric Functions (CCSS.Math.Content.HSF-TF.B.5)
Standard: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
These lessons cover trigonometric functions and amplitude as well as graphing trigonometric functions using amplitude and periodicity.
Inverses of Trigonometric Functions (CCSS.Math.Content.HSF-TF.B.6)
Standard: Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
Use these lessons to discuss inverses of trigonometric functions, and how to find the domain and range of trigonometric functions and their inverses.
Trigonometric Equations (CCSS.Math.Content.HSF-TF.B.7)
Standard: 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.
These lessons show students how to solve trigonometric functions using inverses and include a lesson on problem solving using trigonometric equations.
Trig & Pythagorean Identities (CCSS.Math.Content.HSF-TF.C.8)
Standard: 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 of the angle.
These lessons include a list of basic trigonometric identities and explain how to use the Pythagorean identity.
Addition & Subtraction Formulas (CCSS.Math.Content.HSF-TF.C.9)
Standard: Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
Show students the addition and subtraction formulas for sine, cosine and tangent as well as how to problem solve using these formulas with these lessons.
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