Table of Content

Introduction

1. Intro to Parabolas
2. First and Second Differences

Equation Forms

1. Factored Form
2. Standard Form
3. Vertex Form

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

1. Axis of Symmetry
2. Optimal Value
3. Transformations
4. Step Pattern
5. X-int or zeroes
6. Finding an equation for a parabola

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

1. X-int or zeroes

2. Optimal value

3. Axis symmetry

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

2. axis of symmetry
3. Optimal value
4. Using completing the sq to turn into vertex
5. Factoring
6. Common
7. Simple
8. Complex
9. Perfect Squares
10. Difference of squares

Parabolas

Parabolas are a curve on a graph, you can use parabolas to graph a distance from a soccer ball that is kicked, or even throwing a baseball.

Important terms to remember, these are what make up a Parabola

Axis of Symmetry: It's in the middle of your parabola. It helps to find your vertex.

Vertex Point: This is the curve of the parabola. It's the direction of where it faces.

Optimal Value: This is the value of your vertex.

First and Second Difference

If the first differences are the same it'll create a linear relationship. If the first differences aren't the same but the second differences are it's a quadratic relationship.

Below is a chart of an example of the first and second differences.

Types of equations

There are three different forms you'll need to know.

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

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

Standard form: ax2+bx+c=0

Vertex form

Using the vertex form is one of the equations to graph a parabola. Below are the transformations and key words, and step pattern.

Ex. y= -4(x-2)^2 +6

Vertical Reflection: If it's a - the parabola will open downwards. If it's positive it will open upwards.

Vertical Stretch: This effects the step pattern instead of doing the basic step pattern routine it will double instead look below.

Horizontal Translation: This moves the parabola left or right. It's also the part of the vertex.

Vertical Translation: This moves the parabola up or down. It's also a part of the vertex.

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.

How to find 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

By setting y=0 it helped us find the x-intercepts

Step Pattern

A normal parabola would go. (over one up 1)(over 2 up 4).

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.

Zeros

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

Math 1234