# Quadratic Relationship

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## The three equations that you use in quadratics

Vertex form y=a(x-h)2+k

Factored form a(x-r)(x-s)

standard form y=ax2+bx+c

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

## Axis of symmetry

Axis of symmetry cuts the parabola in half. It is the X coordinates. A parabola is the U shape on the graph. There are two different formulas to find the axis of symmetry. If your equation is y = (x-h)^2+k then it is in vertex form. If your equation is y= ax^2+bx+c then it is in standard form. To find the axis of symmetry you need to find the two X-intercepts add them together and divide them by 2 giving you the axis of symmetry.

## Optimal value

Optimal value has maximum value and a minimum value. Maximum value is when the parabola is opening down. Minimum value is when the parabola is opening up. The optimal value is always the Y-intercepts.

## Transformations

Transformation is basically the movement of the graph. some ways the equation can show the movement of the parabola is the horizontal translation,vertical translation,vertical stretch. vertical stretch is basically how much you times the step table by each time, horizontal translation is basically the movement of the parabola left or right, and vertical stretch is the movement of the parabola up or down.

## X-intercepts

To find the x-intercept you have to set y=0.

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

y=3(x+1)^2-108

0=3(x+1)^2-108

+108 +108

108=3(x+1)^2

divide 108 and 3 by 3

36=(x+1)^2

square root 36 and it will cancel out the squared because what you do to one side of the equation you have to do to the other.

6=(x+1)

-1 -1

5=x

This is how you find the x-intercept by putting y=0

## Step pattern

This is basically the way a normal parabola would go. (over one up 1)(over 2 up 4).

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

## x-intercepts (r and s)

The equation for factored form is y= a(x-r)(x-s). With the R and S given you already have x but you have to change the sign to the opposite given. For example: y= (x-6)(x+7) you are going to switch the negative to and positive and positive to a negative (x-6) x=6 (x+7) x= -7. This is one rule you need to do.

## Axis of symmetry (x=(r+s)/2)

To find the axis of symmetry you need to add R and S and then divide them by 2. Thats how you will find the axis of symmetry for factored form.

## Optimal value (sub in)

To find the maximum or minimum value you need to look before the brackets. If the number is positive or negative. If it is a negative the parabola is going to open down and if it is a positive it is going to open up.

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

## Zeroes

In standard form the equation is y =ax^2+bx+c. You will have to make y=0 and then factor it to find x.

## Axis of symmetry

To find the axis of symmetry you first need to find your zeroes. Then you add your x's and then divide them by 2. This gives you the axis of symmetry that cuts the parabola in half.

## Completing the square

You have to change standard form to vertex form.

y= (-4x^2-8x)-1

(b/2)^2

(8/2)^2 = 16

y= (-4x^2-8x+16-16)-1

y= (-4x^2-8x+16)-16-1

y= (-4x^2-8x+16)-17

y= (-2x+4)^2-17

This is one example of turning standard form into vertex form

## How to find x and y intercepts for vertex form

Math 1234