# Quadratic Relationships

### By: Caleb

## What Is a Quadratic Relation?

A quadratic formula is also sometimes called a second degree formula.

There are THREE quadratic equations:

**Vertex Form****Factored Form****Standard Form**

## How does it apply to everyday life?

## An Example of a Parabola

## Unit #1: Vertex Form

## Summary of Graphing Vertex Form

**How to Solve Vertex Form**

**The Placement of the Parabola**

The k and the h in the vertex form determine where the parabola will be on the chart. The** k** represents the vertical translation of the parabola and the h represents the horizontal translation of the parabola.If the k is positive the vertex of the parabola will be on the upper part of the graph and if it the k is negative, the vertex will appear on the bottom half of the graph. If the **h** in the formula is positive that means the parabola will be on the left side of the graph, however if it is negative it will be placed on the right side of the graph.

The **a** determines which way the parabola opens. If the parabola opens downwards, that means that the a is negative. However, if the a is positive, that means the parabola will open upwards

**Step Pattern**

The step pattern is a good way to find out how to graph a parabola. To do the step pattern you must:

· Start at the vertex and go to the next point on the right.

· Identify the rise and the run from the vertex to that point.

· Write them down.

· Then identify the rise and run to the third point.

· Write them down.

· Keep doing so, until you find a pattern in the run. Then do it on the other side to get the parabola.

**Vertex=(h,k)**

**Axis of Symmetry=(x=h)**

**Optimal Value (y=k)**

## Lets Solve a Word Problem

## Unit #2: Factored Form: Y= A(x-r)(x-s)

## Learning Goals

You should also try to be able to make a parabola on a graph using factored form.

## Summary of Factored Form

**The meaning of R, A and S**

The R and S of the equation are actually the two X intercepts of the parabola.

The A of the equation determines whether the parabola opens up or down and if it is stretched or compressed. If it opens upwards, the A must be positive. If it opens downwards, the A will have a negative value. If the A has a value that is greater than 1 or less than -1, the parabola will be stretched vertically. However, if the A has a value that is in between 1 and -1, the parabola will be vertically compressed.

**Different ways of Factoring**

There are 5 different types of factoring, they are:

**Greatest Common Factor: Monomial Factoring**

When we do monomial factoring we factor out the greatest common factor

5x+10y

=5(x+2y)

**Simple Factoring: Simple Trinomial Factoring**

To factor a simple trinomial we need 2 numbers that multiply to get the last number, but add up to get the middle number as shown below.

x²+9+20

=(x+5)(x+4)

**Complex Factoring: Complex Trinomial Factoring**

To solve complex factoring you need to multiply the lead coefficient with the last coefficient. Once you get that number, you must find two numbers that has a product of that number and the sum of the middle number in the trinomial. Once you get those two numbers, replace the middle term with them. After that, just do common factoring.

2x2 – 3x – 35

**2x2 -10x +7x -35**

**2x(x-5) + 7(x-5)**

**(2x+7)(x-5)**

**Difference of Squares**

(x-2)^2 or (x-2)(x-2)

** Perfect Square**

x2+12x+36

(x+6)^2 or (x+6)(x+6)

## Here is an in depth video about factoring trinomials

**Axis of Symmetry/Optimal Value**

There is an equation to find the axis of symmetry using factored form. To get the axis of symmetry you must add up the R and S values and divide them by 2. The equation looks like this: x=r+s/2. Once you solve this, sub this value into the X to get the optimal value or the vertex.

__For example:__

## Finding the y intercept

For example:

## Here is a word problem using factoring!!!

## Unit#3: Standard Form: Y=ax² +bx+c

## Learning Goals

## What do the variables mean?

**in the equation gives you the direction of the opening of the parabola. If it is negative the parabola will open downwards. If the a is positive, however, the parabola will open upwards instead. The value of a will also determine if the parabola will be vertically stretched or compressed. The**

__a__**in the equation is the y intercept. If the C is negative the y intercept will be on the bottom half of the graph. If the C is positive, it will instead be on the top half of the graph.**

__c__## The Quadratic Formula

## When do you use the quadratic formula?

For Example:

## Completing the square to get Vertex Form

## Here are some word problems for standard form!!!

*h=t²-4t+7,*where

*h*represents the height above the ground in metres and

*t*shows the time in seconds.

a) What is the maximum height of the soccer ball?

b) At what time does the soccer ball reach its maximum height?

c) How tall is the building?

## Reflection

When learning vertex form, I learned the characteristics of a parabola like the vertex and y and x intercepts. During the vertex form unit I also learned how create a parabola, name parabola transformations and solve word problems based on quadratics.During the factored form unit, I found out how to expand and simplify quadratic expressions. I also learned different ways of factoring such as complex trinomial factoring, difference of squares, binomial common factoring, perfect square trinomial and factoring by grouping as well as learning to solve factoring word problems. The standard form unit taught me how to find the zeros from the discriminant and how to use the quadratic formula.

This unit has really strengthened my knowledge of graphing, lines and parabolas. Throughout this unit my ability to do word problems has been strengthened and I find graphing much easier. I now also can relate parabolas to real life. I really enjoyed the quadratics unit because of all the new concepts that I have learned.