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
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
Quadratic Formula
Example:
Word Problems
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.