I need a full explanation on how to use quadratic functions in graphing form, and also notes would be great.

The term quadratic comes from the word quadrate meaning square or rectangular. Similarly, one of the definitions of the term quadratic is a square. In an algebraic sense, the definition of something quadratic involves the square and no higher power of an unknown quantity; second degree. So, for our purposes, we will be working with quadratic equations which mean that the highest degree we'll be encountering is a square. Normally, we see the standard quadratic equation written as the sum of three terms set equal to zero. Simply, the three terms include one that has an x2, one has an x, and one term is "by itself" with no x2 or x.

Thus, the standardized form of a quadratic equation is ax2 + bx + c = 0, where "a" does not equal 0. Note that if a = 0, the x2 term would disappear and we would have a linear equation!

[As a review, if the highest degree in an equation is 1, meaning that the x-term is x1 or in the form ax + by = c or y = mx + b, the equation is always linear.]

What about a quadratic equation? What are the characteristics of a quadratic function? Well, if we look at the simplest case when a = 1, and b = c = 0, we get the equation y = 1x2 or y = x2. We talked a little bit about this graph when we were talking about the Vertical Line Test. We said that the graph of y = x2 was a function because it passed the vertical line test.

Let's graph the equation again. Remember, if you are not sure how to start graphing an equation, you can always substitute any value you want for x, solve for y, and plot the corresponding coordinates. So, let's try substituting values in for x and solving for y as depicted in the chart below.

x

y = x2

y = x2

(x, y)

-3

(-3) 2

9

(-3, 9)

-2

(-2) 2

4

(-2, 4)

-1

(-1) 2

1

(-1, 1)

0

(0) 2

0

(0, 0)

1

(1) 2

1

(1, 1)

2

(2) 2

4

(2, 4)

3

(3) 2

9

(3, 9)