Algebra Tutorials!
Sunday 16th of June
Calculations with Negative Numbers
Solving Linear Equations
Systems of Linear Equations
Solving Linear Equations Graphically
Algebra Expressions
Evaluating Expressions and Solving Equations
Fraction rules
Factoring Quadratic Trinomials
Multiplying and Dividing Fractions
Dividing Decimals by Whole Numbers
Adding and Subtracting Radicals
Subtracting Fractions
Factoring Polynomials by Grouping
Slopes of Perpendicular Lines
Linear Equations
Roots - Radicals 1
Graph of a Line
Sum of the Roots of a Quadratic
Writing Linear Equations Using Slope and Point
Factoring Trinomials with Leading Coefficient 1
Writing Linear Equations Using Slope and Point
Simplifying Expressions with Negative Exponents
Solving Equations 3
Solving Quadratic Equations
Parent and Family Graphs
Collecting Like Terms
nth Roots
Power of a Quotient Property of Exponents
Adding and Subtracting Fractions
Solving Linear Systems of Equations by Elimination
The Quadratic Formula
Fractions and Mixed Numbers
Solving Rational Equations
Multiplying Special Binomials
Rounding Numbers
Factoring by Grouping
Polar Form of a Complex Number
Solving Quadratic Equations
Simplifying Complex Fractions
Common Logs
Operations on Signed Numbers
Multiplying Fractions in General
Dividing Polynomials
Higher Degrees and Variable Exponents
Solving Quadratic Inequalities with a Sign Graph
Writing a Rational Expression in Lowest Terms
Solving Quadratic Inequalities with a Sign Graph
Solving Linear Equations
The Square of a Binomial
Properties of Negative Exponents
Inverse Functions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with One Log Term
Combining Operations
The Ellipse
Straight Lines
Graphing Inequalities in Two Variables
Solving Trigonometric Equations
Adding and Subtracting Fractions
Simple Trinomials as Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing Fractions 2
Rational Numbers
Difference of Two Squares
Factoring Polynomials by Grouping
Solving Equations That Contain Rational Expressions
Solving Quadratic Equations
Dividing and Subtracting Rational Expressions
Square Roots and Real Numbers
Order of Operations
Solving Nonlinear Equations by Substitution
The Distance and Midpoint Formulas
Linear Equations
Graphing Using x- and y- Intercepts
Properties of Exponents
Solving Quadratic Equations
Solving One-Step Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic Inequality with Two Solutions
Operations on Radicals
Factoring a Difference of Two Squares
Straight Lines
Solving Quadratic Equations by Factoring
Graphing Logarithmic Functions
Simplifying Expressions Involving Variables
Adding Integers
Factoring Completely General Quadratic Trinomials
Using Patterns to Multiply Two Binomials
Adding and Subtracting Rational Expressions With Unlike Denominators
Rational Exponents
Horizontal and Vertical Lines
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Solving Quadratic Equations

Completing the Square

The Vertex Form of a Quadratic Function

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.

 What the Vertex Form of a Quadratic can tell you about the graph

The vertex form of a quadratic function:

y = a  · (x - h) 2 + k,

also tells you whether the graph of the quadratic is smiling or frowning. To check, simply look at the value of a, as you would if the equation had been written in standard form. If the value of a is positive then the quadratic is smiling and if the value of a is negative then the quadratic will be frowning.

The vertex form of a quadratic equation can also tell you about the location of the highest point (on a frowning quadratic) or the lowest point (on a smiling quadratic – see Figure 1 on the next page). This point (the highest point on a frowning quadratic or the lowest point on a smiling quadratic) is called the vertex.

The x-coordinate of the vertex is the number h that appears inside the parentheses of the vertex form and the y-coordinate of the vertex is the number k that appears outside the parentheses in the vertex form.

Figure 1: (a) In this quadratic, a = -1 and the shape of the graph is a “frown.” The vertex in this case is the highest point on the graph. (b) In this quadratic a = 0.5 and the shape of the graph is a “smile.” The vertex in this case is the lowest point on the graph.

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