About This Chapter
Working with Quadratic Functions - Chapter Summary and Learning Objectives
Now that you've been introduced to quadratics in the previous chapter, use this one to further explore the relationship between these types of equations and their graphs. You can also try out advanced methods for factoring and solving quadratic equations - regardless of whether they have a leading coefficient of one or higher - along with techniques for graphing systems of quadratic inequalities used to model data sets or chart projectile motion. Watch the video lessons in this chapter and learn how to:
- Write the equation of a parabola in its various forms
- Factor quadratics and convert a polynomial into a perfect square
- Solve quadratic equations with real and complex numbers
- Graph quadratic inequalities and systems of quadratic inequalities
|What Is a Parabola?||Get some examples of parabolas in everyday life and learn how to identify their maximum height, vertex, axis of symmetry and intercepts.|
|Parabolas in Standard, Intercept, and Vertex Form||View examples of quadratic equations written in standard, intercept and vertex forms. Check out their graphs.|
|How to Factor Quadratic Equations: FOIL in Reverse||Review the processes involved in multiplying binomials. Learn how to factor equations by performing this process in reverse.|
|Factoring Quadratic Equations: Polynomial Problems with a Non-1 Leading Coefficient||Explore advanced techniques for factoring polynomials with leading coefficients higher than one.|
|How to Complete the Square||Find out what it takes to turn a polynomial into a perfect square binomial.|
|Completing the Square Practice Problems||Get extra practice converting the standard form of a quadratic equation into the vertex form. Convert a trinomial into a binomial by completing the square.|
|How to Solve a Quadratic Equation by Factoring||Learn how factoring and the zero product property can be used to solve quadratic equations.|
|How to Solve Quadratics with Complex Numbers as the Solution||Discover the uses of the discriminate to determine whether the solution to an equation contains real or complex numbers. Practice solving quadratics that include complex numbers.|
|Graphing & Solving Quadratic Inequalities: Examples, Lesson & Quiz||Get the definition of a quadratic inequality. Learn how to solve and graph these types of equations.|
|Graphing a System of Quadratic Inequalities: Examples, Lesson & Quiz||Discover methods for graphing quadratic inequalities as well as systems of quadratic inequalities.|
|Applying Quadratic Functions to Motion Under Gravity and Simple Optimization Problems||Understand how quadratic functions can be used to model projectile motion and solve optimization problems.|
|Using Quadratic Functions to Model a Given Data Set or Situation||Explore the uses of quadratic functions to model data sets.|
1. What is a Parabola?
A parabola is the U shape that we get when we graph a quadratic equation. We actually see parabolas all over the place in real life. In this lesson, learn where, and the correct vocab to use when talking about them.
2. Parabolas in Standard, Intercept, and Vertex Form
By rearranging a quadratic equation, you can end up with an infinite number of ways to express the same thing. Learn about the three main forms of a quadratic and the pros and cons of each.
3. How to Factor Quadratic Equations: FOIL in Reverse
So, you know how to multiply binomials with the FOIL method, but can you do it backwards? That's exactly what factoring is, and it can be pretty tricky. Check out this lesson to learn a method that will allow you to factor quadratic trinomials with a leading coefficient of 1.
4. Factoring Quadratic Equations: Polynomial Problems with a Non-1 Leading Coefficient
Once you get good at factoring quadratics with 1x squared in the front of the expression, it's time to try ones with numbers other than 1. It will be the same general idea, but there are a few extra steps to learn. Do that here!
5. How to Complete the Square
Completing the square can help you learn where the maximum or minimum of a parabola is. If you're running a business and trying to make some money, it might be a good idea to know how to do this! Find out what I'm talking about here.
6. Completing the Square Practice Problems
Completing the square is one of the most confusing things you'll be asked to do in an algebra class. Once you get the general idea, it's best to get in there and actually do a few practice problems to make sure you understand the process. Do that here!
7. How to Solve a Quadratic Equation by Factoring
If your favorite video game, 'Furious Fowls,' gave you the quadratic equation for each shot you made, would you be able to solve the equation to make sure every one hit its target? If not, you will after watching this video!
8. How to Solve Quadratics with Complex Numbers as the Solution
When you solve a quadratic equation with the quadratic formula and get a negative on the inside of the square root, what do you do? The short answer is that you use an imaginary number. For the longer, more helpful answer, check out this lesson!
9. Graphing & Solving Quadratic Inequalities: Examples & Process
When you finish watching this video lesson, you will be able to graph and solve your own quadratic inequality. Learn what steps you need to take and what to watch for.
10. Graphing a System of Quadratic Inequalities: Examples & Process
What is a system of quadratic inequalities? What are the steps to graphing them? Find the answers to these questions by watching this video lesson. When you finish, you will be able to graph these yourself.
11. Applying Quadratic Functions to Motion Under Gravity & Simple Optimization Problems
Quadratic functions are very useful in the real world. Watch this video lesson and learn how these functions are used to model real world events such as a falling ball.
12. Using Quadratic Functions to Model a Given Data Set or Situation
Watch this video lesson to learn how quadratic functions are used to model situations and data gathered from real world scenarios. See how our function fits our data and situation well.
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Other chapters within the Precalculus: High School course
- Working With Inequalities
- Absolute Value Equations
- Working with Complex Numbers
- Introduction to Quadratics
- Basics of Polynomial Functions
- Working with Higher-Degree Polynomials
- Graphing Piecewise Functions
- Understanding Function Operations
- Graph Symmetry
- Graphing with Functions
- Rate of Change
- Rational Functions & Difference Quotients
- Rational Expressions and Function Graphs
- Exponential Functions & Logarithmic Functions
- Using Trigonometric Functions
- Trigonometric Graphs
- Trigonometric Applications
- Solving Trigonometric Identities
- Vectors, Matrices and Determinants
- Mathematical Sequences and Series
- Sets in Algebra
- Analytic Geometry and Conic Sections
- Polar Coordinates and Parameterizations