About This Chapter
Quadratic Functions & Polynomials - Chapter Summary
Use the lessons in this chapter to find out what, exactly, a parabola is. Learn how to manipulate the direction of these functions by writing quadratic equations in the standard, intercept and vertex forms, and discover methods for maintaining the correct order of operations when multiplying binomials and factoring quadratic expressions.
Once you've mastered the basics, learn the steps for factoring a trinomial into a square binomial. You can also practice using the quadratic formula and study techniques for solving quadratic equations that aren't in standard form. Additional lessons show you how to graph a quadratic equation to locate the x- and y-intercepts. There are also tips for finding the maxima and minima of a quadratic function, determining the end behavior of a polynomial and graphing monomials. A lesson on factoring quadratic functions to find zeros rounds out the chapter, which is designed to teach you the following main topics:
- Factoring quadratic equations
- Using the quadratic formula
- Graphing monomials and polynomials
Our video lessons put the expertise of experienced instructors at your disposal. Watch them model the steps for solving quadratic equations and use the corresponding lesson transcripts to peruse the examples and explanations provided. If you'd like to try your hand at solving some of these problems, there are also quizzes to help you assess your comprehension of each lesson's content.
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. 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!
7. How to Use the Quadratic Formula to Solve a Quadratic Equation
When solving a quadratic equation by factoring doesn't work, the quadratic formula is here to save the day. Learn what it is and how to use it in this lesson.
8. How to Solve Quadratics That Are Not in Standard Form
It isn't always the case that your equation will be set up nicely for you to solve. In this lesson, learn how to factor or use the quadratic formula to solve quadratic equations, even when they are not in standard form.
9. How to Solve a Quadratic Equation by Graphing
This lesson will show us how to solve a quadratic equation by graphing. Through definitions, illustrations, and applications, we will learn the steps involved in this process.
10. Finding Minima & Maxima: Problems & Explanation
One of the most important practical uses of higher mathematics is finding minima and maxima. This lesson will describe different ways to determine the maxima and minima of a function and give some real world examples.
11. Finding Zeroes of Functions
The zero of a function is the point (x,y) on which the graph of the function intersects with the x-axis. The y value of these points will always be equal to zero. There can be 0, 1, or more than one zero for a function.
12. Using Linear & Quadratic Functions to Problem Solve
Linear and non-linear functions can be used to model and find solutions to a plethora of problems. Examples of non-linear functions may include exponential and quadratic functions.
13. Polynomial Functions: Properties and Factoring
Everything from projectile motion to trigonometric functions can be described by polynomials. Review factoring, polynomials and quadratic functions in this lesson.
14. Understanding Basic Polynomial Graphs
This lesson will cover understanding basic polynomial graphs. The lesson focuses on how exponents and leading coefficients alter the behavior of the graphs.
15. Analyzing Graphs of Polynomial Functions
This lesson will explain the graph of a polynomial function by identifying properties including end behavior, real and non-real zeros, odd and even degree, and relative maxima or minima. We will then sketch a graph using this information.
16. Using Rational & Complex Zeros to Write Polynomial Equations
In this lesson, you will learn how to write a polynomial function from its given zeros. You will learn how to follow a process that converts zeros into factors and then factors into polynomial functions.
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Other chapters within the TExES Mathematics 7-12 (235): Practice & Study Guide course
- About the TExES Math 7-12 Exam
- Real Numbers
- Mathematical Models
- Complex Numbers & the Complex Plane
- Number Theory
- Number Patterns
- Functions and Graphs
- Linear Functions
- Evaluating Piecewise & Composite Functions
- Rational and Radical Functions
- Inequalities and Absolute Values
- Exponentials & Logs
- The Unit Circle
- Trigonometric Functions
- Using a Scientific Calculator for Calculus
- Understanding Limits in Math
- Understanding Rate of Change
- Calculating Derivatives of Functions
- Derivatives and Graphs
- Optimization in Calculus
- Definite Integrals and Sums
- Integration Applications in Calculus
- Working with Measurement
- Finding Volume, Area & Perimeter
- Introduction to Proofs and Constructions
- Congruence and Similarity
- Real World Shapes
- Coordinate Geometry
- Understanding Transformations in Math
- Conic Sections
- Understanding Vectors
- Measuring & Displaying Data
- Data Distribution Overview
- Sampling in Statistics
- Distribution & Inference in Statistics
- Inference About a Mean
- Regression and Correlation
- Finding Probability
- Probability Distributions and Statistical Inference
- Experiments and Surveys
- Mathematical Process & Perspectives
- Teaching Strategies & Activities for the Math Classroom
- Differentiated Instructional Strategies for the Math Classroom
- Using Student Assessments in the Math Classroom
- TExES Mathematics 7-12 Flashcards