Quadratics

By: Sajithan Sethukavalar

What are Quadratics

Quadratics is an algebra similar to linear systems but deals with curved lines. The type of quadratic you will be learning about on my poster is graphing by vertex form. The vertex form equation is y= a(x-h)²+k.

Linear vs. Non-Linear

In grade 9 you probably learned about linear equations (straight lines) using the formula y=mx+b. After you learn about quadratics (curved lines) using a completely different equation y=a(x-h)²+k. To establish whether the line is linear or not you look a the first and second differences. First you have to find the differences of values under the y column. If the first differences are increasing or decreasing at a constant rate, that means it is linear. If it's not constant, then you have a non-linear equation, to find out if it is quadratic you have to take the first differences and find the differences of them, as shown in the example to the right.
Big image

Vertex Form

y=a(x-h)²+k

In quadratics vertex form is only one the forms of a quadratic equation. On this site you will be learning about axis of symmetry, vertex, the step pattern as learned above, how to find the x intercept, transformations and how to graph a parabola using the vertex form equation.

Step Pattern

The step pattern determines the path of the parabola. The standard step pattern for the base graph is is over one, up one, then over one up 3 then over one up 5. If the "a" value is a different number from one then you will multiply the pattern (1,3,5...) by the value of "a".

The "x" and new "y" values are the points making the parabola.

Big image

Terms

Optimal Value (Min and Max Value)

  • It is the value of the y in the vertex point
  • Minimum value is when graph opens up
  • Maximum value is when graph opens down

Axis of Symmetry

  • It is a line divides the parabola in half making it the same on each side
  • It is the "h" in the equation and is always opposite of what it appears in the bracket.

Vertex (H,K)

  • The vertex is the highest point of parabola
  • It is the point where the graph changes direction
  • The vertex can be found in the equation in the form of "h" and "k"

Tranformations

"a"

The "a" value determines the opening and vertical stretch or compression of the parabola

Shape:

  • If the value of "a" is more than one then the parabola is vertically stretched
  • If the value of "a" is less than one then the parabola is vertically compressed

Orientation:

  • If the value of "a" is negative the parabola opens down
  • If the value of "a" is positive the parabola opens up

"h"

The "h" value determines if the parabola moves right or left

Orientation:

  • If the "h" value is negative the parabola moves to the right
  • If the "h" value is positive the parabola moves to the left

"k"

The "k" value determine whether the parabola is moved up or down.

Orientation:

  • If the "k" value is positive the parabola is moved up.
  • If the "k" value is negative the parabola is moved down

Graphing

There are two different ways to graph using the vertex form
  • The way I like to use is to graph a parabola using the vertex form, you can start by finding and plotting the vertex. Then you multiply using the step pattern and plot the points, then connect them.
  • The other way is mapping notation.It is mainly going from y=x² to y=a(x-h)² +k. This is done by making a chart with "x" values then subbing it into the equation making points as shown below.
Big image
Graphing Quadratic Functions Using Transformations