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Functions

Functions. Linear and Quadratic. Overview. In this module you will be introduced to functions, as well as learning about a class of functions called linear and quadratic . You will also be given access to some tools that are necessary to understand functions.

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Functions

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  1. Functions Linear and Quadratic

  2. Overview In this module you will be introduced to functions, as well as learning about a class of functions called linear and quadratic. You will also be given access to some tools that are necessary to understand functions. Linear functions describe the relationship between the independent (x) variable and the dependent (y) variable that when graphed creates a line. Quadratic functions describe the relationship between the independent variable (x) and the dependent variable (y) that when graphed create a parabola. These functions are polynomials with a second degree. There is a green star on each of the topic slides. Clicking on this start will return you to the beginning of the presentation. Each webpage is hyperlinked to open right from this presentation as well.

  3. Topic: Overview of Functions • What is a function? • How do you determine if something is a function? • Domain of a Function • Range of a Function • Function Notation • Build Functions Tools of the quadratic equations • Radicals • Adding and Subtracting Radicals • Multiplying and Dividing Radicals • Simplifying Radicals • Complex Numbers Useful websites: • Graphing Calculator Online • Printable Graph Paper

  4. Topics - Linear Functions • Slope • Intercepts (x and y) • Horizontal vs. Vertical Lines - Which is a function? • Point-Slope Form • Slope-Intercept Form • Standard Form • Parallel Lines • Perpendicular Lines • Linear Inequalities

  5. Topics - Quadratic Functions • Quadratic Equation - Definitions • Exploring the quadratic equation • What is a quadratic function? • What does a quadratic function look like? • Focus & Directrix • Parabola • Graphing Parabolas • Graphing with a graphing calculator • Graphing quadratics in vertex form • Solving Quadratic functions • Factoring • Completing the Square • Quadratic Formula • Complex Quadratics • Quadratic Inequalities

  6. What is a Function? Meat-A-Morphosis • Meat-A-Morphosis • You Tube Video • Jason Ermer • This quick video demonstrates a ‘Function Machine’ using a real life example, as well as demonstrating mathematical examples.

  7. Function: Yes or No? Recognizing a Function • Testing if a relationship is a function • Khan Academy • This webpage gives a quick, precise description of how to test is a relation is a function. • There are many videos, all listed on the left, that show different ways to test if a relation is a function.

  8. Domain vs. Range Domain vs Range • Graphing, Domain and Range of Functions • Regents Test Prep • This webpage explains the definition of domain and range, specifically in relationship to graphing.

  9. Function Notation Function Notation • Functions & Function Notation • Khan Academy • This webpage on Khan Academy provides another description of functions. This description of functions also goes into detail about how to use function notation. • Please note that there are several different videos (all listed on the left side of the page). • Also, please remember that f(x) is function notation.

  10. Building Functions Building Functions • Mathematical Models, Building Functions • School Tube • This video goes into detail showing numerous examples of building functions. • The initial examples show how to build functions from values. • The remaining examples show how to build functions from other functions.

  11. Graphing Calculator - Ad free Online Graphing Calculator Desmos Calculator (Graphing) • Desmos Calculator • This online graphing calculator allows (majority) of the features of a hand held graphing calculator • If you wish to save your functions, you can create a username. • Graphing Calculator • This graphing calculator does not allow points to be plotted in conjunction with the equation • There are no ads

  12. Printable Graph Paper Worksheet Works • Printable Graph Paper • Worksheet Works • This website allows customizable graph paper to be created, printed, and used for educational purposes. • Once you are on this website, type graph paper in the search box to select the type of graph paper you want.

  13. Simplifying Radicals Multiplying & Dividing Radicals Adding and Subtracting Radicals Square Roots and Radicals REMEMBER: CLICK BACK TO GET BACK TO THIS PRESENTATION ONCE YOU HAVE VISITED THE RESOURCE!

  14. Adding and Subtracting Radicals Adding and Subtracting Radicals Khan Academy • This resource provides information on how to add and subtract radicals.

  15. Multiplying and Dividing Radicals Multiplying and Dividing Radicals Multiplying and Dividing Radicals: The Math Page • This resource provides examples on how to multiply and divide radicals. • There are links to further resources and examples from this site.

  16. Simplifying Radicals Square Roots & Radicals Simplifying Radicals • Simplifying Radicals • Math Meeting • This website provides a video that explains how to analyze and simplify radicals, including square roots • Simplifying Radicals: Introduction and Simplification • Purple Math • This site provides an overview to simplifying radicals. It has a video as well as reading material to support the topic. • There is a link to a self test as well as a link to extra problems for practice

  17. Complex Numbers Complex Numbers • Complex Numbers • Khan Academy • This webpage has a detailed explanation of complex numbers. There are also many video showing examples. • There are interactive problems to try (all listed on the left hand side of the page).

  18. Slope Slope of a Line • Slope of a Line • The Math Open Reference Project • This interactive website provides a description of slope, how to find slope, and what a lines looks like with a positive, negative, undefined, or zero slope. • There is a practice portion where the user can create different lines to discover how slope changes.

  19. Intercepts X and Y Intercepts • X and Y Intercepts • Khan Academy • This webpage includes a video explanation of x and y intercepts. • There are also examples shown of how to graph using the intercepts, as well as how to find the intercepts given an equation in different forms.

  20. Horizontal & Vertical Lines Horizontal & Vertical Lines • Horizontal & Vertical Lines • The Open Math Reference • This interactive website provides an explanation of horizontal and vertical lines. • There is also an interactive applet that allows the user to explore with vertical and horizontal lines.

  21. Point-Slope Form of a Line Point Slope Form • Point - Slope Form of a line • Khan Academy • This webpage describes from where point slope form of a line develops, shows examples of how to exact the point and slope from an equation, as well as how to write an equation of a line in point slope form • There are interactive examples with answers.

  22. Slope-Intercept Form of a Line Slope-Intercept Form of a Line • Equation of a Line - Slope Intercept Form • The Math Open Reference Project • This interactive website outlines the slope-intercept form of a line. There is also a portion that allows the user to see how the slope and intercept affect the look of the line. • y = mx + b

  23. Standard Form of a Line Standard Form of a Line • Standard Form Equation of a Line • Math Warehouse • This webpage describes the standard form of line including a video showing an example. • There are also examples of how to write and manipulate the equation of a line, including steps and answers.

  24. Parallel Lines Parallel Lines • Parallel Lines • The Math Open Reference Project • This resource provides a definition of parallel lines as well as an interactive applet to allow for the user to manipulate points and viewing their effect on the slope. • There are 2 example problems given. • This site provides hints and guidance for the user on how to use the page to further their understanding. • There are links at the bottom of this page to other coordinate geometry topics.

  25. Perpendicular Lines Perpendicular Lines • Perpendicular Line • The Math Open Reference Project • This resource provides a definition of perpendicular lines as well as an interactive applet to allow for the user to manipulate points and viewing their effect on the slope. • There are 2 example problems given. • This site provides hints and guidance for the user on how to use the page to further their understanding. • There are links at the bottom of this page to other coordinate geometry topics.

  26. Practice Problems Linear Inequalities Linear Inequalities Math Planet • This resource provides an information on solving linear inequalities. • There are links on the right hand side to take the learner to solving compound inequalities, systems of inequalities, solving absolute value inequalities, and linear inequalities in 2 variables. Shodor.org • This resource provides an interactive applet to practice graphing inequalities.

  27. Quadratic Equation Quadratic Equation • Quadratic Equation: Wolfram Alpha • This site provides a definition of the quadratic equation including links to terms such as complex numbers, quadratic formulas and completing the square

  28. Quadratic Equation Quadratic Equation • Math is Fun: Quadratic Equation • This site provides explanations of the quadratic equation, explains ways to solve the quadratic equation. • The site includes interactive applets for exploration as well as an equation solver app. • A song to remember the quadratic formula is on this site • Additional practice questions are at the bottom of this site.

  29. Quadratic Function :What is a Quadratic Function • Virtual Nerd Video • This video provides an overview of what a quadratic function is. • This resource provides links to other tutorials such as: • What is a function • Definition of a quadratic function (parabola) • Simplifying expressions • What is standard form of a quadratic?

  30. Parabola: Focus and Directrix Parabola Overview • Parabola: Focus and Directrix Video • The Khan Academy • This Khan Academy Video provides an overview of the focus and directrix of a parabola.

  31. Graphing Parabolas Graphing Parabolas • How to Graph a Parabola • NY State Regents • This site provides 2 examples of how to graph a parabola including information on an equation in quadratic form, axis of symmetry and maximum and minimum. • There is a link at the top of the site to additional topics: Algebra Index and Topic Index • There is a link at the top of the site to the Regents Exam Prep Center • There is a link at the bottom of the site page to Graphing Calculator Help.

  32. Using the TI-83/84 to Graph a Parabola Graphing with a Calculator • Using the TI-83/84 to graph a Parabola • This website provides instructions on how to use the TI-83/84 graphing calculator to graph a parabola. It includes information about the setting the window as well as typing in the quadratic function.

  33. Graphing Parabolas Graphing Parabolas • Khan Academy Video: Graphing Parabolas in vertex form • This video provides an overview of how to graph parabolas in vertex form.

  34. Completing the Square Completing the Square • Khan Academy Video: Completing the Square • Video demonstrating how to use completing the square on any quadratic equation.

  35. Free Math Help: Quadratic Formula This site provides instructions for using the quadratic formula. There is an interactive problem solving tool available on this site. Quadratic Formula Quadratic Formula Quadratic Formula • Khan Video: Quadratic Formula • This video provides an overview of how to use the quadratic formula to solve quadratic functions

  36. Hard Case Examples Factoring Solve Quadratics by Factoring • Purple Math • Solving quadratic functions by factoring • This site provides examples of how to solve quadratics by factoring. There is a video as well as step by step examples for the reader to follow. • There are embedded links for the if the reader needs additional hints or help with the example problems. • Purple Math • Solving hard cases : solving quadratics by factoring • Hard case examples; including what to do when the leading coefficient is -1

  37. Complex Numbers Complex Numbers • Math is Fun: Complex Numbers • This site is provides an overview of complex numbers including some practice problems.

  38. Quadratic Inequalities Quadratic Inequalities Quadratic Inequalities: Math is Fun • This resource provides some colorful examples of how to work with quadratic inequalities. • There are real world examples provided as well as links at the bottom to more topics(solving inequalities, inequalities, and other algebra topics)

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