Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Relating Equations and Graphs for Quadratic Functions

Quadratic Equations

A quadratic equation is an equation that can be simplified to follow the pattern:

y = a · x 2 + b · x + c,

where the letter x represents the input, the letter y represents the value of the output and the letters a, b and c are all numbers. Sometimes the numbers a, b and c are referred to as coefficients.

For an equation to be a quadratic equation, the number that multiplies the x 2, that is, the number a cannot be equal to zero. It is okay for either or both of the numbers b and c to be equal to zero, but a cannot be equal to zero.

Sometimes you might see a quadratic equation written in a format like:

y = x 2 + 2 · x + 3,

and wonder if this is okay, as there does not seem to be a value of a multiplying in front of the x 2. In this situation everything is okay, because the value of the number a in this situation is a = 1. We could write 1 · x 2 in the quadratic equation, but because a factor of “1” does not change the value of anything, the factor of “1” in 1 · x 2 is usually just left out.

Other Ways of Writing Quadratic Equations

The format for a quadratic equation given above,

y = a · x 2 + b · x + c,

where the letter x represents the input, the letter y represents the value of the output and the letters a, b and c are all numbers, is called standard form.

Other ways of writing the equations for quadratic functions include vertex form,

y = a · (x - h) 2 + k,

where the letter x represents the value of the input, the letter y represents the value of the output and the letters a, h and k all represent numbers. Just as in standard form, in vertex form the number a cannot be equal to zero. Converting a quadratic equation to vertex form is often quite helpful as it allows you to determine exactly where the graph of the quadratic equation reaches its “low point” or “high point” very easily. Every single quadratic formula can be converted to vertex form. The process for doing this conversion is called completing the square.

 The third common way of writing the formula for a quadratic function is called factored form in which the quadratic function is written as a product of two factors,

y = a · (x - c) · (x - d).

Many (but not all) quadratic functions can be written in factored form. The quadratic functions that can be written in factored form are the ones whose graphs touch the x-axis in at least one place. If the graph of the quadratic function never touches the x-axis then the formula of the quadratic function cannot be converted to factored form.

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