Table Of Contents

Vertex Form
  • Vertex Form Equation
  • Axis of Symmetry, Optimal Value
  • Solving for (a) Value
  • Zeroes
  • Graphing Vertex Form

Factored Form

  • Expanding and Simplifying Binomials
  • Simple Trinomial Factoring
  • Complex Trinomial Factoring
  • Common Factoring
  • Factoring by Grouping
  • Perfect Square
  • Difference of Sqaures


Quadratics is the study of curved lines which is also known as a Parabola. An example of a quadratic is when you kick a soccer ball. The soccer ball goes up then down and makes a curved line and quadratics can help you find the highest point when the soccer ball was in the air and even how long it took to get in reach the highest point.

Vertex Form

The written equation for vertex form is y=a(x-h)²+k.

The (a) value of the equation determines if the parabola is a vertical stretch or compressed. If the (a) value is less than one, the parabola will be compressed. If the (a) value is one or greater than the parabola is vertically stretched. The (-h) value and (k) value of the equation are the vertex coordinates of the parabola. The vertex is the highest point in the parabola. The (-h) value is the x coordinate the (k) value is the y coordinate. The vertex also gives the axis of symmetry and optimal value.

Axis of Symmetry/ Optimal Value

The axis of symmetry is the (-h) value, it is the x coordinate in the parabola that divides it in half equally.The optimal value is the (k) value, it is the y coordinate in the parabola that is is the highest point in the parabola. The AOL and Optimal Value create the Parabola's Vertex.

Solving for ''a'' Value

When solving for the ''a'' value, you have to plug in a point from the parabola into the equation that has the (h) value and (k) value. The ''a'' value is not always given to us, in order to solve for it we have to isolate ''a''.


When we have the (a) value given to us in the vertex form equation, y=a(x-h)²+k . Then the (a) value given to us multiples the step pattern.

Vertical Stretch or Compressed?

The (a) value either makes the parabola stretch vertically or compress.

If the (a) value is less than one it compresses, if its one or more it vertically stretches.

Direction of Opening

The direction of opening is the term used to say whether the parabola is opening up or down.

The parabola will have a direction of opening facing up if the (a) value is positive.

If the (a) value is negative the parabola will have a direction of opening facing down. There is a reflection on the x axis when the (a) value is negative.

Graphing Vertex Form

Graphing in Vertex is simple we have the information from Vertex form to know where everything goes.

Example; y=(x+4)^2

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3.2 Graphing from Vertex Form

Factored Form

Graphing Factored Form

The (a) value still multiplies the step pattern or reflects the parabola, just like how it did in vertex form. In vertex form we are give the vertex in the equation, well in factored form we are given the two x intercepts. The (r) and (s) values are the x intercepts of the parabola. One thing to remember is, if the (r) or (s) value is negative. When we graph the parabola the x intercept will not be negative it will be positive. If the (r) or (s) value is positive then the x intercepts will be negative.

For Example : (x-4)(x+2)

Notice how the x-ints are switched.

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Expanding and Simplifying

Expanding factored forms will give us our simplified trinomial or binomial.To expand we can use the algebra tiles method or the F.O.I.L method. F.O.I.L stands for First, Outer, Inner and Last. We first multiple the first terms in the factored form then we multiple the outer terms, inner terms and finally the last terms After we're done expanding we simplify by collecting like terms.
3.6 Expanding

Simple/ Complex Trinomial Factoring

Common Factoring

When factoring an equation, sometimes you will have to see if the trinomial or binomial has a common factor. Below is a video on how to common factor.
Commom Factoring

Factor by Grouping

Perfect Square

Difference of Squares / Expanding

A difference of squares is always a binomial. Both terms in the binomial have a square root but the second term is always a negative. Here is a video where I teach you this method.
Difference of Squares