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FunctionsLinear and Quadratic
OverviewIn 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.
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
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
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
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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.
What is a Function?
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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.
Function: Yes or No?
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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.
Domain vs. Range
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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.
Function Notation
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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.
Building Functions
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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
Online Graphing Calculator
Graphing Calculator - Ad free
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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.
Printable Graph Paper
Square Roots and RadicalsSimplifying Radicals
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Multiplying & Dividing Radicals
Adding and Subtracting Radicals
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Adding and Subtracting Radicals
Khan Academy• This resource provides information on how to add and
subtract radicals.
Adding and Subtracting Radicals
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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.
Multiplying and Dividing Radicals
Multiplying and Dividing Radicals
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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
Square Roots & Radicals
Simplifying Radicals
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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).
Complex Numbers
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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.
Slope
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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.
Intercepts
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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.
Horizontal & Vertical Lines
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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.
Point-Slope Form of a Line
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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
Slope-Intercept Form of a Line
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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.
Standard Form of a Line
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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.
Parallel Lines
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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.
Perpendicular Lines
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Linear InequalitiesMath 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.
Linear Inequalities
Practice Problems
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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
Quadratic Equation
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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.
Quadratic Equation
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: 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?
Quadratic Function
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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.
Parabola: Focus and Directrix
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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.
Graphing Parabolas
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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.
Using the TI-83/84 to Graph a Parabola
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Graphing Parabolas
• Khan Academy Video: Graphing Parabolas in vertex form• This video provides an overview of how to graph parabolas in
vertex form.
Graphing Parabolas
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Completing the Square
• Khan Academy Video: Completing the Square• Video demonstrating how to use completing the square on
any quadratic equation.
Completing the Square
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Quadratic Formula
• Khan Video: Quadratic Formula• This video provides an overview of how to use the quadratic
formula to solve quadratic functions
Quadratic Formula
• 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
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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
Factoring
Hard Case Examples
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Complex Numbers
• Math is Fun: Complex Numbers• This site is provides an overview of complex numbers including some practice problems.
Complex Numbers
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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)
Quadratic Inequalities
Quadratic Inequalities
