Algebra Tutorials!
   
Home
About Us




TUTORIALS:

Absolute Values
Solving Two-Step Equations Algebraically
Multiplying Monomials
Factoring Trinomials
Solving Quadratic Equations
Power Functions and Transformations
Composition of Functions
Rational Inequalities
Equations of Lines
Graphing Logarithmic Functions
Elimination Using Multiplication
Multiplying Large Numbers
Multiplying by 11
Graphing Absolute Value Inequalities
Polynomials
The Discriminant
Reducing Numerical Fractions to Simplest Form
Addition of Algebraic Fractions
Graphing Inequalities in Two Variables
Adding and Subtracting Rational Expressions with Unlike Denominators
Multiplying Binomials
Graphing Linear Inequalities
Properties of Numbers and Definitions
Factoring Trinomials
Relatively Prime Numbers
Point
Inequalities
Rotating a Hyperbola
Writing Algebraic Expressions
Quadratic and Power Inequalities
Solving Quadratic Equations by Completing the Square
BEDMAS & Fractions
Solving Absolute Value Equations
Writing Linear Equations in Slope-Intercept Form
Adding and Subtracting Rational Expressions with Different Denominators
Reducing Rational Expressions
Solving Absolute Value Equations
Equations of a Line - Slope-intercept form
Adding and Subtracting Rational Expressions with Unlike Denominators
Solving Equations with a Fractional Exponent
Simple Trinomials as Products of Binomials
Equivalent Fractions
Multiplying Polynomials
Slope
Graphing Equations in Three Variables
Properties of Exponents
Graphing Linear Inequalities
Solving Cubic Equations by Factoring
Adding and Subtracting Fractions
Multiplying Whole Numbers
Straight Lines
Solving Absolute Value Equations
Solving Nonlinear Equations
Factoring Polynomials by Finding the Greatest Common Factor
Logarithms
Algebraic Expressions Containing Radicals 1
Addition Property of Equality
Three special types of lines
Quadratic Inequalities That Cannot Be Factored
Adding and Subtracting Fractions
Coordinate System
Solving Equations
Factoring Polynomials
Solving Quadratic Equations
Multiplying Radical Expressions
Solving Quadratic Equations Using the Square Root Property
The Slope of a Line
Square Roots
Adding Polynomials
Arithmetic with Positive and Negative Numbers
Solving Equations
Powers and Roots of Complex Numbers
Adding, Subtracting and Finding Least Common Denominators
What the Factored Form of a Quadratic can tell you about the graph
Plotting a Point
Solving Equations with Variables on Each Side
Finding the GCF of a Set of Monomials
Completing the Square
Solving Equations with Radicals and Exponents
Solving Systems of Equations By Substitution
Adding and Subtracting Rational Expressions
Percents
Laws of Exponents and Dividing Monomials
Factoring Special Quadratic Polynomials
Radicals
Solving Quadratic Equations by Completing the Square
Reducing Numerical Fractions to Simplest Form
Factoring Trinomials
Writing Decimals as Fractions
Using the Rules of Exponents
Evaluating the Quadratic Formula
Rationalizing the Denominator
Multiplication by 429
Writing Linear Equations in Point-Slope Form
Multiplying Radicals
Dividing Polynomials by Monomials
Factoring Trinomials
Introduction to Fractions
Square Roots
   
Try the Free Math Solver or Scroll down to Tutorials!

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


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

Like standard and vertex forms, the factored form of a quadratic function:

y = a · (x - c) · (x - d)

can also tell you whether the given quadratic has a graph that “smiles” or “frowns.” As with the standard and vertex forms the key again is the sign (+ or -) of the number a. If a is positive the graph “smiles” and if a is negative the graph “frowns.” Sometimes you will see an example of a factored form that does not appear to have a value of a, such as:

y = (x -1) · (x + 2).

In this case, the value of a is equal to one (which is positive) and the graph of the quadratic will smile.

The factored form of a quadratic equation also tells you where the x-intercepts (sometimes called the roots or zeros of the quadratic function are located). The xintercepts of the equation are the x-values that will make y = 0.

For example, the x-intercepts of the quadratic:

y = (x -1) · (x + 2)

are x = 1 and x = -2 as plugging either of these two values into the quadratic equation will make y equal to zero.

Example

Figure 1 shows the graph of a quadratic function. Find the equation of this quadratic and express your answer in both factored and standard forms.

Figure 1: Find the formula of this quadratic function.

Solution

Figure 1 clearly shows both the of the x-intercepts of the quadratic function so it will be easiest to find the factored form of the quadratic first and then convert this to standard form by FOILing.

The x-intercepts of the quadratic shown in Figure 4 are located at x = 0 and x = 4. This means that the factored form of the quadratic function must look something like this:

y = a · (x - 0) · (x - 4) = a · x · (x - 4) .

The factored form must have a factor of (x - 0), or more simply a factor of just x, to ensure that when you plug in x = 0 the value of y will be equal to zero. The factored form must also have a factor of (x - 4) to ensure that when you plug in x = 4 the value of y will be equal to zero.

To determine the numerical value of a you can plug in the x- and y-coordinates of any other point on the quadratic graph (i.e. any point other than one of the x-intercepts) and solve for a. Figure 1 shows that the point (2, 2) lies on the graph, so you can plug in x = 2 and y = 2 into the factored form. Doing this:

2 = a · 2 · (2 - 4)

2 = a · (-4)

So, the equation of the quadratic function from Figure 1 (written in factored form) is:

To convert this equation to standard form, you can expand by FOILing and then simplify (if necessary). Doing this:

(Expand by FOILing)

(Multiply through by - beware of “-” signs)

The formula for the quadratic shown in Figure 1 (expressed in standard form) is:

 

Copyrights © 2005-2017
Tuesday 12th of December