# Quadratics

### Made By: Bahar Wahab

## Inroduction

__Quadratics?__

When it comes to quadratics it may not be the easiest unit, but with much practice and patients you will definitely get the hang of it. The name Quadratic comes from "quad" meaning square, because the variable gets squared, like x^2.

__When Are Quadratics Used? __

You may have not noticed, but quadratics surround us everyday without anyone giving much attention. For example, throwing a ball and it landing on the ground, the arch of a rainbow, the water that comes out of a fountain, etc.

Some real life examples of a quadratic would be:

## Types of Equations

- Factored Form: y= a(x-r)(x-s)
- Standard Form: y= ax^2+bx+c
- Vertex Form: y= a(x-h)^2+k

## Second Differences

## Vertex Form

__Parabolas and Transformation__

__Graphing using transformations__

__Find the zeros__

__Word problems__

__Finding an equation given the vertex and 2 points__

__Graphing using step pattern__

__Reflection__

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

## Learning Goals

For many individuals math may not be a subject which they can easily excel in, but putting up a few learning goals would get you on the right path. A few learning goals I set up for my self would be:

- Labeling a parabola.
- Find the zero's in vertex form.

## Introduction to Parabolas

A parabola is what is created when a quadratic relation is graphed. The vertex form for a quadratic relation is y= a(x-h)^2+k. Parabolas have key features such as:

- Vertex: The point where the axis of symmetry meets the parabola and it is the lowest or highest point of the parabola depending on its direction.
- Optimal Value: the y value of the vertex/ the lowest or highest point of the parabola.
- Axis of Symmetry: a vertical line that divides the parabola into two equal halves.
- Y-intercept: the point where the parabola intercepts the y axis.
- X-intercept or Zeroes : the point(s) where y=0.

## Transformation

- The a in the equation stretches or compresses the parabola. If the the a is positive the parabola will open up but if the a is negative the parabola will open down/reflect about the x axis. When describing this you would say there is a vertical stretch or compression by a factor of whatever number a is.

- The h is the horizontal translation. If h is negative the parabola will translate right and if it is positive the parabola will translate left. When describing this you would say there is a horizontal translation to the left or right by whatever number h is.

- The k is the vertical translation. When k is negative the parabola will move down and when it is positive the parabola will move up. When describing this you would say there is a vertical translation up or down by whatever number k is.

- (h,k) are the vertex. in the equation when stating the vertex make sure to switch the x sign. Example: y=2(3-2)^2-9 vertex: (2,-9)

## Findng the zeros in vertex form

## Vertex Form Word Problems

## Economic Problem

## Finding an equation given the vertex and two points

## Reflection

## Factored Form

__Zeros__

__Multiplying Binomials__

__Binomial common factoring__

__Monomial factoring__

__Factoring by grouping__

__Factoring simple trinomials__

__Decomposition__

__Special Cases__

__Graphing factored form__

__Factored form word problems__

__Reflection on the unit__

r and s are the x-intercepts/roots/zeros

(r,0) (s,0)

The value of a will tell you the shape and direction of the parabola

## lEARNING gOALS

- Multiplying binomials by using foil.
- Factoring by grouping.

## Zeros/X-intercepts/Roots

Example:

y=(x+6)(x-3)

0=(x+6)(x-3)

x+6=0

x=-6

x-3=0

x=3

The x intercepts are: (-6,0) (3,0)

## Multiplying Binomials: Expanding

**F**irst term

**O**utside term

**I**nside term

**L**ast term

__There are also special products:__

Perfect Square:

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

__There are also special products:__

Perfect Square:

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

## Binomial Common Factoring

Example: 2y(w+3)+2x(w+3)

The 2 binomials are the same they are both (w+3) (common factor) so they will become one. Now if you divide both terms by (w+3) you will be left with (2y+2x) making your factored equation: (w+3)(2y+2x)

For more help watch the video below

## Monomial Commmon Factors

Example: 7x+21y

These 2 have a common factor of 7 so 7 is the number we will divide by making our new factored equation: 7(x+3y)

Too see another example and for more help watch the video below.

## Factoring By Grouping

Example: xy-5y+4x-12

Group together xy and 4y because they both have a common factor of y and group together 3x and 12 because they have a common factor of 3. The equation now becomes:

y(x-4)+3(x-4)

=(x-4)(y+3)

For more help and examples watch the video below.

## Factoring Simple Trinomials

x^2+bx+c to (x+r)(x+s). The method that will be used is called product and sum where c is your product (rs=c) and b is your sum (r+s=b). In this method you need to find 2 numbers whose product will be equal to c and whose sum will be equal to b. There are a few rules when doing product and sum which are:

- When b and c are both positive, r and s will also both be positive
- When b is negative and c is positive both r and s will be negative
- When c is negative either the r or s is negative

Example:

x^2+5x+6

Product=6 Sum=5

What 2 numbers will give me a product of 6 and a sum of 5?

2x3=6

2+3=5

so the 2 numbers are 2 and 3

The new equation will be:

(x+2)(x+3)

## Decomposition (Complex Trinomials)

For more help watch the video below

## Special Cases

## Graphing Factored form

- Find the x intercepts/roots/zeros
- Use the x intercepts to find the axis of symmetry (the x in the vertex)
- Plug the axis of symmetry (x) into the original equation to find y
- Plot on your graph!

## Factored Form Word Problem

## Measurement Problems

## Reflection

## Standard Form

__Completing the square__

__The quadratic formula__

__Discriminant__

__Graphing standard form__

__Word problems__

## Learning Goals

- Finding the x-intercepts and vertex of a standard form equation to be able to graph it.
- Solving for number problems.

## Completing The Square

**y=ax^2+bx+c** to **y=a(x-h)^2+k**

Here's how to do it:

**1.** You are going to be starting off with the equation in the form **y=ax^2+bx+c**. Place ax^2+bx in to brackets so it looks like **y=(ax^2+bx)+c**.

**2.** Within the brackets if the a value is not equal to one common factor so its just **ax^2** and the number you are common factoring is outside of the brackets.

**3.** Now take the b value, divide by 2 and then square root it and the number that you get add it into the brackets and also subtract it to make it equal. It will looks like

**y=(ax^2+bx+#-#)+c**.

**4.** Move the number that you are subtracting outside the bracket (remember if you are taking a number out of brackets it has to be multiplied by the number in front of the bracket) and add or subtract it with c. It will look like

**y=(ax^2+bx+#)+#+c**.

**5.** Use product and sum to find the number that will be put into the bracket and the equation will now be in vertex form:

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

Example:

**1.** Our equation is in standard form to begin with:** y=ax^**2**+bx+c**.

**2.** We want to put it into vertex form: **y=a(x-h)**2**+k**.

**3**. We can convert to vertex form by completing the square on the right hand side.

**4. **36 is the value for 'c' that we found to make the right hand side a perfect square trinomial.

**5. **Our perfect square trinomial factors into two identical binomials, **(x+6)•(x+6)**.

**6.** The vertex of an equation in vertex form is **(h,k)**, which for our equation is **(-6,-4)**.

## Quadratic Formula

**If the number under the square root is negative the quadratic has no solution.**All you need to do is plug the a,b, and c value in to the formula and solve.

## Discriminant

If the discriminate is less than **0** that means there are no solutions/x-intercepts/roots.

If the discriminate is equal to** 0 **that means there is 1 solution.

If the discriminate is over **0** that means that there are 2 solutions/x-intercepts/roots.

## Graphing Standard Form

- Find the x-intercepts (quadratic formula)
- Find the vertex (completing the square)