# Quadratics

### Quadratic Relationships

## Vertex Form y=a(x-h)²+k

A quadratic equation in vertex form is written y=a(x-h)²+k. When you graph a quadratic equation in vertex form, you have to write the vertex which is given in the equation, then look at the step pattern shown by the value of a. In this equation the vertex is represented by h and k. The h being the x value and the k being the y value. 'h' may be written as negative or positive, the sign would be reversed when writing it in the vertex. Occasionally you may be given the vertex, and a point, which means that you would have to solve for the 'a' value to complete the equation and graph.

- vertex is (h,k)
*(The maximum or minimum point on the graph. It is the point where the graph changes direction)* y=k is the optimal value

- x=h is the axis of symmetry
*(divides parabola in half)*

**The value of (a)--> determines the orientation and shape of the parabola**

__Orientation__

If a>0, the parabola opens up

If a<0, the parabola opens down

__Shape__

If -1<a<1, the parabola is vertically compressed

If a>1 or a<-1, the parabola is vertically stretched

**The value of (k) --> determines the vertical position of the parabola**

If k>0, the vertex moves up by k units

If k<0, the vertex moves down by k units

**The value of (h) --> determines the horizontal position of the parabola**

If h>0, the vertex moves to the right h units

If h<0, the vertex moves to the left h units

**Example:**

## Factored Form y=a(x-r)(x-s)

A quadratic equation in factored form is written as y=a(x-r)(x-s). In this equation, to figure out the x-intercepts you put each factor = 0, then solve for x. Another way that you can figure out the x-intercepts is by reversing the signs for both 'r' and 's'. If 'r' and 's' are positive the x-intercepts would be negative and if they are negative then the x-intercepts would be positive. After you have figured out both of the x-intercepts, add both the x values and divide them by 2 (r+s/2). This will lead you to knowing x=h which is also known as the axis of symmetry. Once you have done this, you have to find out the 'k' which is the optimal value. To find out the optimal value you substitute the 'h' (axis of symmetry) value into the equation, the value of y is k.

**In general if y=a(x-r)(x-s)**

__The zeros are found by setting each "factor" equal to zero so:__

x-r=0 means x=r and x-5=0 means x=5

__The axis of symmetry__

- is the midpoint of the two zeros so x=r+s/2

__The optimal Value__

-is found by subbing the axis of symmetry value into the equation

## Common Factoring

## Simple Trinomial

## Complex Trinomial

## Perfect Squares & Difference of Squares

## Unit Connections

__Equations and Graphing__

All three equations can be graphed

__Completing the Square and Vertex From:__

When completing the square you take a equation from standard from to vertex form. By Completing the square you end up with a quadratic equation from standard form to vertex form and this is the main reason why they both connect.