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Basic Quadratic Terminology
A quadratic expression is one where the largest power for the variable is 2. Some examples of quadratic expression are shown below.
A quadratic equation is an equation where the largest power for the variable is 2. Remember that an equation has an equal sign in it. Some examples of quadratic equations are:
Quadratic equations can be solved in order to find the roots of the equation. Roots are also called zeros or x–intercepts if the graph crosses the x–axis. The roots of a quadratic equation simply tell what values of x will make the equation true.

A quadratic function is a function where the largest power for the variable is 2. A function usually takes the form of y = or f(x) =.

So what are the key differences between an expression, an equation, and a function?
  • A quadratic expression is usually just something that can be simplified or factored. You cannot solve an expression for a variable. You can only manipulate the terms you are given.
  • A quadratic equation is given to you so that you can solve it for the variable.
  • A quadratic function is given to you so that you can graph it.

Examples
The graph of a quadratic function is called a parabola. A parabola can either have a “u” shape or an “n” shape depending on the number in front of the term. If the number in front of x2 is positive, the parabola will open up; if the number in front of x2 is negative, the parabola will open down.
Example
In which direction will the graph of open?
What is your answer?
 
Example
In which direction the graph of open?
What is your answer?
 

Regardless of whether the parabola opens up or down, all parabolas will have a vertex. If the parabola opens up, the vertex is the minimum point or the place where the graph bottoms out. If the parabola opens down, the vertex is the maximum point or the place where the graph reaches its peak.

Quadratic functions can be written in one of two ways, depending on what you are trying to do with the equation. The two key pieces for graphing a quadratic equation are having the roots and the vertex. Along with the vertex and the roots, you can plot several other points to get a complete graph of a quadratic. If the roots are imaginary, plotting additional points is required to get an accurate graph.

Examples
For each question, determine whether the information given represents a quadratic expression, a quadratic equation, or a quadratic function.
Example
#1: y = -3x2 - 4
What is your answer?
 
Example
#2: -3x2 + x - 1 = 0
What is your answer?
 
Example
#3: -3x2 + 4x
What is your answer?
 
Example
#4: 6x2 = 5x
What is your answer?
 
Example
#5: 3x2 = 5x - 7x2 + 1
What is your answer?
 
Example
#6: y = ½(x-2)2 - 4
What is your answer?
 



S Taylor

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