### By Nisha Sabharwal

The name Quadratic comes from "quad" meaning square, because the variable gets squared. Quadratic equations make a smooth curve.

## Reflection:

In the beginning of this unit when we started with quadratics 1 I was struggling with what to do but when we started quadratics 2 I got a better understanding and once I understood what to do in quadratics 2, quadratics 3 was much easier. This improvement didn't however show on my tests because I was still somewhat unclear with quadratics 1 and that made me unsure on my answers because they would be related on the tests.

1. Vertex form: y=a(x-h)2+k
2. Standard form: y=ax2+bx+c
3. Factored form: y=a(x-r)(x-s)

First Differences

• Optimal value
• Axis of Symmetry (A.O.S)
• x-intercepts/zeros
• Exact roots
• Approximate roots
• Discriminants

## Standard Form:

y=ax2+bx+c

Zeros:

Quadratic Formula (-b+- square root of b2-4ac/2a)

5x2-7x+2=0

• 5 is the a term 7 is the b term and 2 is the c term
7 +- square root of (-7)2-4(5)(2)/2(5)

• since 7 was a negative in the equation it gets changed to a positive

7 +- square root of 9/10

7 + square root of 9/10

=1

7 - square root of 9/10

=0.4

the zeros are (1, 0.4)

Axis of Symmetry:

-b/2a

-(-7)/2(9)

7/10

=0.7

Optimal Value:

sub in 0.7 into the equation

y=5(0.7)2-7(0.7)+2

y=0

Completing the Square:

y=x2+8x+4

y=(x2+8x=42-42)+4

y=(x2+8x+42)-42+4

y=(x+4)2-12

• (2 represents squared)

Factoring

## Finding the Vertex:

The vertex is made up of the h and k value in the equation.

Example: y=(x+6)2+12

Vertex: (-6,12)

*If the h value is positive in the bracket, when plotting your vertex it turns negative. And if in the bracket it is negative, when plotting the h value turns positive (i.e. y=2(x-8)2-9, vertex= (8,-9)). The k value stays the same when plotting

## A.O.S. and Optimal Value:

• A.O.S=the h value of vertex equation, it is also the point the divides the parabola in half.
• Optimal value=the k value of vertex equation

Example: y=-(x+5)2+9

• A.O.S= 5
• Optimal Value= 9

## Note:

• if first difference the same=linear relation
• if second difference same=quadratic relation (parabola)
• if no difference the same=neither

y=a(x-h)2+k

*2 stands for squared

a= if the parabolas opens up or down (negative=opens down and positive=opens up), compressed or stretched (if a decimal or fraction=compressed, if a whole #=stretched).

h=how many units the parabola horizontally moves (-#=moves to the left, +#=moves to the right).

k=how many units the parabola vertically moves (-#=moves down, +#=moves up).

Vertex= (h,k) the maxima or minima of the curve(parabola)

## Transformations

a= if the parabolas opens up or down (negative=opens down and positive=opens up), compressed or stretched (if a decimal or fraction=compressed, if a whole #=stretched).

h=how many units the parabola horizontally moves (-#=moves to the left, +#=moves to the right).

k=how many units the parabola vertically moves (-#=moves down, +#=moves up).

Vertex= (h,k) the maxima or minima of the curve(parabola)

## Finding Equations in Vertex Form

When working with vertex form, we are usually looking to solve for a. With many problems regarding this form the h,k(vetex),x,and y(another point) values will be given, as you have to do is sub in all the values bring the numbers to one side solve for a. Once get your values write your equation with the h,k,and a values.

## Finding Zeros/x-intercepts and y-intercepts in Vertex Form:

*when finding the y-intercept, always sub x=0

Example: y=-4(x+2)2+16

• y=-4(x+2)2+16
• y=-4(0+2)2+16
• y=-4(2)2+16
• y=-4(4)+16
• y=-16+16
• y=0

*when solving to find the x-intercepts, remember to always sub y=0

• y=-4(x+2)2+16
• 0-16=-4(x+2)2
• -16/-4=(x+2)2
• 4=(x+2)2
• take square root on both sides
• +2 or -2=x+2
• 0 or -4=x

therefore the x intercepts are 0 and -4

## Step Pattern

The step pattern determines where to plot your points in the parabola. To determine you step pattern, you must know what a equals. Once you know your a value then, multiply your a value with 1,3, and 5 to get your step pattern.

## A.O.S

• A.O.S=the h value of vertex equation, it is also the point the divides the parabola in half.
• Optimal value=the k value of vertex equation

Example: y=-(x+5)2+9

• A.O.S= 5
• Optimal Value= 9

Factored form: y=a(x-r)(x-s)

*r and s represent zeros (x-intercepts)

1. find zeros
2. find A.O.S
3. find optimal value

Example:

y=-2(x+4)(x+6)

1. Find zeros:
• x+4=0 x+6=0
• x+4-4=0-4 x+6-6=0-6
• x=-4 x=-6

2. A.O.S= -4+(-6)/2

• = -10/2
• = -5 (x value of vertex)
• 3.Optimal Value (sub -5 as x in original equation)
• y=-2(-5+4)(-5+6)
• y=-2(-1)+(1)
• y=2 (y/k value of vertex)

Vertex: (-5,2)

Vertex form: -2(x+5)2+2

## Difference of Squares

When working with difference of squares, you are multiplying two binomials to get a binomial.

Formula: a2-b2=(a+b)(a-b)

when expanding (a+b)(a-b):

• (a+b)(a-b)
• a2-ab+ab-b2
• a2-b2

## Word Problems

Word Problems