# Quadratic Relations

### Gurjot Parmar

## Contents

- Axis of symmetry

- Optimal Value

- Transformations

- X intercepts

- Step pattern

Factored Form:

- X intercepts

- Axis of symmetry

- Optimal value

Standard Form:

- Zeroes

- Axis of symmetry

- Optimal value

- Completing the square

- Factoring to turn to factored form

- Common
- Simple trinomial
- Complex trinomial
- Perfect squares
- Difference of squares

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

## Transformations

'a' controls whether the parabola opens up or down (if the 'a' value is negative it opens down, if the 'a' value is positive it will open up)

'a' also controls whether the parabola is stretched or compressed

'-h' controls the horizontal shift (left or right)

'k' controls the vertical shift (up or down)

## Step Pattern

The step pattern for y = x^2 :

y = 1^2

y = 1 x 1

y = 1

This means that we move one unit over and one unit up (1,1).

y = x^2

y = 2^2

y = 2 x 2

y = 4

This means that we move 2 units over and 4 units up (2,4)

## Axis of Symmetry

y = a(x -** h**)^2 + k

## Optimal Value

## How to find x-intercepts for vertex form

0 = (x-3)^2 -16

Then bring the -16 to the other side and square root both sides to simplify.

√-16 = √(x-3)^2

4 = x-3

Now add and subtract the (-3) from 4 to find the x intercepts.

4+3= 7

-4+3= -1

This tells us that the x intercepts are 7,0 and -1,0.

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

## How to graph factored form

y = -3(x-4)(x+2)

First you want to find the x intercepts by making the brackets equal to zero.

x-4 = 0 x+2 = 0

x = 4 x = -2

(4,0) (-2,0)

After finding the x intercepts you want to plug use the equation r+s/2 to find the Axis of Symmetry.

= -4+2/2

= -2/2

Axis of symmetry= 1

Now you want to plug in the AOS (1) into the equation to find the optimal value.

y = -3(x-4)(x+2)

y = -3(1-4)(1+2)

y = -3(-3)(3)

y = 27

Optimal Value = 27

Now we know the vertex is (1,27) and the x intercepts and now we just have to graph it.

## Standard Form y = ax^2 + bx + c

## Quadratic Formula

Example:

*= x^*2 + 3*x* – 4

x = -3 + √3^2 -4 (1)(-4)

= -3+ √9+16 / 2

= -3+√25 /2

= -3+5 /2

= 2/2

=1

=-3-5

=-8/2

=-4

X intercepts are -4 and 1.

## Completing the square

y = 2x^2 +8x + 5

y = (2x^2 +8x) + 5

Than you want to factor out the 2. Do not factor the 2.

y = 2 (x^2 +4x) + 5

After that you want to divide the b value by 2 and square it.

y = 2 (x^2 +4x/2) + 5

y = 2 (x^2 +2x^2) + 5

y = 2 (x^2 +4x+4-4) + 5

Now you want to bring the -4 out of the bracket and multiply it with the a value. After multiplying it add it with 5.

y = 2 (x^2 +4x+4) -8+ 5

y = 2 (x^2 +4x+4) -3

Now you factor out the square in the bracket by square rooting everything inside of it.

y = 2 (x^2 +4x+4) -3

y = 2 (x+2) -3

And now you have completed the square and your equation is in vertex form.