# Quadratics

### By: Apinder Sudan

## Table Of Contents

## Intro

-Linear and non-linear

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

-Optimal Value

-AOS x=(r+h)÷2

## Standard Form - y = ax2 + bx + c

-Zeroes

-Optimal Value

-Completing Square

-Factoring

1.Common

2.Simple trinomial

3.Complex trinomial

4.Perfect Square

5.Difference of squares

## Vertex Form: y=a(x-h)2 + k

- Axis of symmetry (x=h)
- Optimal Value (y=k)
- Transformations (translation vertical or horizontal, vertical stretch, reflection
- Step Pattern
- X-intercepts or Zeroes (sub y=0 and solve)

## What is quadratics?

## Linear and non-linear

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

## Zeros/X Intercepts

The X-intercepts are the points or the point at which the parabola intersects the x-axis. A parabola can have either two, one, or zero real x intercepts.

## Axis Of Symmetry (AOS)

Every parabola has an axis of symmetry which is the line that runs down the middle . This line divides the graph into half. The axis of symmetry in factored form uses the formula x=(r+h)÷2. the value of x is the axis of symmetry.

## Optimal Value

To find the optimal value you would need to first find the two x-intercepts and then sub them into the equation (r+h) ÷ 2) and solve.

## Standard Form - y = ax2 + bx + c

## Zeros Formula

## Optimal Value

## Common Factoring

9k^3-27k-145 = 9(k^3 - 3k -18)

## Factoring with a simple trinomial

A simple trinomial is an equation that has three different terms, and the value of a is one, in the standard form equation (y=ax^2+bx+c) to factor you need two brackets, and both brackets need to equal to the equation.

## Difference Of Squares

example:

(a+b)(a-b)

a^2-ab+ab-b^2

this equals to a^2-b^2 because a and b get cancelled out each other.

## Perfect Squares

The first terms square root is 4m and the last terms square root is 3. When you multiply those two you get 12m and multiply that by 2 and you get 24m. So that is a perfect square. Now when you factor it, you should get (4m+3)^2

## Vertex Form

- y=a(x-h)^2+k
- h and k are the points of the vertex so, (h,k)
- The sign given with the h isn't the right sign, you have to bring "h" to the other side of the equal sign to find out what it is, an example is if it (x-8) it would be x-8=0 next you bring the 8 to the other side changing it sign, so x=8
- If "a" is positive the parabola would open upwards, a "U" shape, if "a" was negative then you would have a "n" shaped parabola
- If a<1 the graph widens meaning it is compressed, for example: y=0.5(x-3)^2+7, you would say the parabola is compressed
- If a>1 the graph stretches meaning it would become narrower, for example: y=2(x-2)^2+2, you would say the parabola would have a vertical stretch of 2

## Axis Of Symmetry

-To find your Axis of Symmetry from the Vertex Form you would need to look at the H value.

in the equation y= 3(x+2)^2+5 the H value which is 2 turns into a negative and is your axis of symmetry.

- You would write you axis of symmetry like: x= -2. To find the AOS (axis of symmetry) you would change the sign of what is in the bracket +2 would be -2.

## Optimal Value

-If the k value is negative then the vertex will be below the x axis and if the k value is positive then it will be above the x axis

## Transformations

-y=a(x-h)^2+k is the general equation for the quadratic.

-where a controls if the parabola opens up or down.

-where a controls if the parabola is stretched or compressed.

-where h controls the horizontal shift.

-where k controls the vertical shift.

-All these are used when using transformations.

-If A is a positive number then the parabola will be opening up but if it is a negative it will be opening down.

-If the A value is greater than 1 then it is being stretched, if it is >1 but <0 then it is being compressed, if it is 0 then nothing changes.

-If the H value is increased by a number than the graph will shift up by that many points, but if it is decreased than it will shift down that many points.

-If the K value is increased than the graph shifts right by that many points, but if it is decreased than it shifts left by that many points.

## Step Pattern

-The step pattern is the movements that you follow from the vertex to find the other points of the parabola so that you can connect them and create a parabola.

**The Pattern:**

over 1 up 1

over 2 up 4

over 3 up 9

etc.

-also a key piece of information is that when you are about to use the step pattern you must multiply the up number by your a value so if your a value was 2 then your step pattern would be

over 1 up 1x2

over 2 up 2x2

## X intercepts and Zeroes

y=2(x+2)^2-8

0=2(x+2)^2-8 Sub Y for 0

1. 8=2(x+2)^2 Bring 8 over to the other side

2.8/2=(x+2)^2 Divide both sides by 2

3.4=(x+2)^2 Square both sides by - and +

+-2=x+2 calculate for the X's

-2=x+2

-2-2=x

*-4=x*

2=x+2

2-2=x

__x=0__