# Quadratics Relationships

### By Alicia Persaud

## Table of Contents

__Introduction__

- What are quadratic relationships
- Introduction to parabolas

__Vertex form: y= a(x-h)^2+k__

- Axis of symmetry
- Optimal values
- Transformations
- Step pattern
- Graphing vertex form
- Personal assessment/small reflection

__Factored form: y= a (x-r)(x-s)__

- Zeroes or x-intercepts
- Axis of symmetry
- Optimal value
- Word Problem
- Graphing factored form

__Standard form: y=ax^2+bx+c__

- Using the Quadratic Formula and Discriminants
- Word Problem
- Completing the square to turn to vertex form
- Factoring to turn to factored form
- Graphing standard form
- Personal assessment/small reflection

__Conclusion__

- Connections
- Overall reflection

## What is quadratics?

## Introduction to Parabolas

- Vertex: The maximum or minimum point of the parabola. It is the point where the parabola changes direction.
- Axis of symmetry: Divides parabola in half.
- Optimal value: Minimum or maximum value.
- Y-intercept: When x=0, parabola crosses y-axis.
- X-intercept: When y=0, parabola crosses x-axis.

## Vertex Form

## Vertex Form This is the equation used for vertex form with a simple parabola demonstrating how the a value affects the direction opening. | ## Axis of Symmetry The red line in this diagram represent the axis of symmetry. It demonstrates how the x value of the vertex is by showing that the line passes through that point. | ## Optimal Value The small green point on this diagram represents the optimal value, which in this case, is a maximum point. The arrow demonstrates how the optimal value is determined by indicating what the optimal value is. |

## Vertex Form

## Axis of Symmetry

## Axis of Symmetry

## Optimal Value

## Transformations

- The a value stretches/compresses the parabola, the direction of opening, and the step pattern.
- The -h value translates the parabola horizontally. *RECALL*- if it is negative in the equation, it will be positive on the graph! Therefore you will move it to the right as opposed to the left, and vice versa.
- The k value translates the graph vertically. It multiplies the vertical part of the step pattern.

__step pattern__to find other points. What is the step pattern, you ask?

## Step Pattern

## Graphing Vertex Form

## Personal Assessment and Small Reflection

## Factored Form

## Zeroes or X-intercepts

## Axis of Symmetry

For example,

As mentioned before, you must first add the two x-intercepts:

**(-5)+(3)=-2**

Now, you have to divide by two.

**-2/2=-1**

As shown in the diagram above, the axis of symmetry is -1.

## Optimal Value

## Word Problem

## How to Graph Factored Form

**Step one:**Find the zeroes. Recall- to find the zeroes, you must set y=0, therefore you will have to take the value of the x-intercepts and carry them to the other side of the equation to isolate x.

*This will give you the two zeroes.*

**Step two:** Once you plot the two points on the graph, you must find the axis of symmetry. You learned earlier that to do so, you must add the two x-intercepts and divide that by 2, once you have done that, then you can create your axis of symmetry (h-value of the vertex).

**Step three:** You must now find the optimal value. It was stated earlier that you must substitute in your axis of symmetry to get this answer. After solving the equation, you will then get your optimal value (k-value of the vertex).

Here is a video to further your understanding:

## Standard Form

**ax^2+bx+c=0.**The variables a, b, and c, are known values; a cannot be 0, x is an unknown value. There are 3 methods in regards of finding the solutions. They are:

- Factor the quadratic
- Complete the square
- Use the quadratic formula

## Using Quadratic Formula and Discriminants

## How to Use the Quadratic Formula

Originally, you are given the equation in standard form, which is ax^2+bx+c. The first thing you must do is find the value of a, b, and c, in your equation. The equation is *x*² - 4*x* - 5 = 0

## Word Problem

## Discriminants

A discriminant is the number inside of the square root (b^2-4ac) and it is useful because it helps up determine how many solutions a quadratic equation has.

- If the discriminant is less than 0, there are NO solutions.
- If the discriminant is greater than 0, there are 2 solutions.
- If the discriminant is equal to 0, there is 1 solution.

## Completing the Square to Turn to Vertex Form

__For the equation y=-3x²+24x-27__

Step one: Group like terms (x^2 and x terms) together by putting brackets around them.

## Factoring to Turn to Factored Form

- Common factoring
- Simple trinomials
- Complex Trinomials
- Perfect Squares
- Difference of Squares

__Common Factoring__Common Factoring is used when there there is a number or variable that is common between the terms in a polynomial. When you factor, you are dividing it out of the set of terms, and putting it in front of the brackets. Watch the video below on how to common factor.

__Simple Trinomial__A simple trinomial often looks like so: x^2+bx=c.

When factoring this, we want to find two numbers that multiplies to give you the c term, and adds to give you the b term. Once you find those two terms, you can simply set out two pairs of brackets and distribute the x^2 into them so there is an x in each set. You can then place the two values you found into a set of brackets. Here is an example:

__Complex Trinomials__You must be wondering what's so different about a complex trinomial than a simple trinomial, or what makes it more complex. Well, in a complex trinomial, there is a coefficient in front of the x value that is not 1. Two methods you can use are the decomposition method and the trial and error method.

__Decomposition:__

Step one: Multiply the a and c value, use that answer as you would originally use your c term.

Step two: Just like in a simple trinomial, find two numbers that will both multiply to give you your c value, and add to give you the b value.

Step three: Now, replace your middle term with the values the terms you just got as your answer.

Step four: Common factor! You will know you did it right if you have the same answers left in brackets. Now you just have to put the values that are outside the brackets in a set of brackets, and in the other set you can leave the values that were the same in previous step.

__Trial and Error:__

Trial and error is a method used in a wide variety of cases when it comes to the principles of mathematics. To use the trial and error method, you have to do exactly what it sounds like- try, and end up with an error; you must keep trying until you get it right. Here is a video that shows you how to factor using the trial and error method:

__Perfect Squares and Difference of Squares__1, 4, 9, 16, 25, 36, 49, 64, 81, 100... these are all examples of perfect squares.

This is useful because of something called perfect square trinomials.

Note: a^2+2ab+b^2= (a+b)^2

a^2-2ab+b^2= (a-b)^2.

Difference of squares is a squared number subtracted from another squared number. It can be expressed as (a+b)(a-b)=a^2-a^2.

Watch this video on how to factor perfect square trinomials and difference of squares: