# Quadratics Website

### By: Jay Patel

## Welcome

## Table Of Contents

- How To Find The Axis Of Symmetry

- Optimal Value

- Transformations

- X-Intercepts and Zeros

- Step Pattern

- How to Graph a parabola in Vertex Form

- How To Change Forms

- Vertex Form to Standard Form

- Vertex Form to Factored Form

2. Factored Form

- How to Find Zeros and X-Intercepts

- Axis Of Symmetry

- Optimal Value

- How to Graph a parabola in Factored Form

- How to Change Forms

- Factored Form to Vertex Form

- Factored Form to Standard Form

3. Standard Form

- Zeroes

- Axis of Symmetry

- Optimal Value

- Discriminants

- Completing the Square to go from Standard Form to Vertex Form

- Common Factoring

- Simple Trinomial Factoring

- Complex Trinomial factoring

- Perfect Squares

- Difference of Squares

- How to change Forms

- Standard Form to Factored Form

- Standard Form to Vertex Form

4. Word Problems

5. Reflection

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

## Axis Of Symmetry

## What Is the Axis Of Symmetry ?

## How do you find the Axis Of Symmetry?

## Optimal Value: (y = K)

## Transformations:

When looking at the formula of y= a(x-h)2+k, each variable shows something important to help you graph the equation.

## The 'a' Variable , shows you if the direction of opening is up or down

i. A negative variable a means the parabola faces up. For example, if the a value is -3, then the parabola faces up

ii. A positive variable a means the parabola faces down. For example, if the a value is +3 then the parabola faces down

iii. If the a value is bigger than 1 (a > 1) or bigger than -1 ( a < -1), then it is called a vertical stretch

iv. If the a value is lower than one but then bigger than -1 ( -1 < a <+1) then it is called a vertical compression

## The 'h' Variable, shows if the vertex moves right or left

i. A positive h value means that the vertex moves Right. For example, if the value is +8 then the vertex would more 8 spaces to the right

ii. A negative h value means that the vertex moves left. For example if the value is -8 then the vertex would move 8 spaces to the left

## The 'k' Variable , shows you if the vertex moves up or down

i. A positive k, shows you how many spaces up the vertex goes. For example, if k= 6 then the vertex would move 6 spaces

ii. A negative k, shows how many spaces down the vertex goes. For example, if k=-6 then the vertex would move down 6 spaces

## How To Find The Vertex

## Here's what questions to answer when you are asked to describe an equation

i. Find out if the parabola opens up or down

ii. Is the parabola vertically compressed or vertically stretched?

iii. How many units the vertex has gone up or down

iv. How many units the vertex has gone to the left or right

v. Find Out what the vertex is

## X-Intercepts and Zeros

## Step Pattern:

## How to Graph an equation in Vertex Form

## How to change forms

## Vertex Form to Standard Form

## Vertex Form to Factored Form

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

## Zeros Or X- Intercepts ( r and s)

When looking at the factored form, the two brackets are actually the two x-intercepts of the parabola. Finding the two intercepts of the parabola is quite easy! First, you take the terms in both brackets and make them equal to zero. Then, you would isolate for x, and you will get your two x-intercepts! After that you would take your x-intercepts and turn then into co-ordinates, by making the x value the x-intercept, and making the y value 0.

## Axis Of Symmetry: ( x= (r+s)/2)

The Axis of Symmetry is the invisible line that divides the parabola equally in half, and the Axis of Symmetry helps in finding the optimal value. To find the Axis of Symmetry you would add the two x-intercepts you found when finding x-intercepts and then divide the answer you get by 2 to get the A.O.S.

## Optimal Value (Sub In)

To find the Optimal Value, you would take the Axis of Symmetry you just found and replace it with the x variable in the equation. After that by getting the “y” value you just found, you would take that number and your A.O.S. that you found and get your vertex!

## How To Graph using the Factored Form

## How to change forms

## Factored Form To Vertex Form

## Factored Form to Standard Form

## Standard Form ( y= ax2 + bx + c )

## Quadratic Formula

## Axis Of Symmetry

## Optimal Value

## Examples Of Finding the Zeroes, Axis of Symmetry and Optimal Value

## Discriminants

## Types Of Factoring

## Common Factoring

## Simple Trinomial Factoring

## Examples

## Complex Trinomial Factoring

## Examples

## Perfect Squares Factoring

## Difference Of Squares

## How to change forms

## Standard Form to Factored Form

## Standard form To Vertex Form

## Word Problems

## Example #1

A soccer ball is kicked into the air and follows a parabolic path described by the equation, h=-2(t-4)2 +15.

a. When does the soccer ball reach its maximum height?

b. What is the maximum height reached?

c. What is the height of the soccer ball at 6 seconds?

For 1b) All I had to do was look at the k value because that showed me the y value of the vertex, which shows how big the height is

For 1c) All I had to do was replace t with 6 and then solve

## Example #2

The height of a rock thrown from a walkway over a lagoon can be approximated by the formula h=-5t2+20t+60, where “t” is the time in seconds, and “h” is the height, in meters.

a. Write the above formula in factored form

b. When will the rock hit the water?

For 2b) I found the x-intercepts

## Example #3

A concert promoter models the profit from her next concert, P dollars, by the equation P=-11(t-55)2+10571, where t dollars is the cost per ticket.

a. What price should she charge to maximize her profit? What is the maximum profit?

b. What would be her profit if the tickets were free?

c. How much should she charge per ticket to break-even?

For 3b) To do this, you would replace the variable t with 0 and then solve

For 3c) You would change p to 0 and then solve.

## Example #4

A flare is released into the air following the path, h=-5(t-6)2+182, where h is the height in meters and t is the time in seconds.

a. What was the flare’s maximum height?

b. What was the flare’s initial height?

c. How long was the flare in the air?

For 4b) I put the variable t equal to 0 and then solved it

For 4c) I put the height to zero, since the height would be zero when the rocket hits the ground and then solved

## Example #5

## Example #6

Ms. Semler has told them:

- At a cost of $3 per ticket, there would be 700 students at the dance

- If the prices will go up, less students will come

- Every time the ticket price goes up by $1, the will lose 70 students.

What price should SAC charge to maximize revenue?