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

The 3 main types of equations used in quadratics is:
  • 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

Using first and second differences can make it easy to figure out whether the equation is quadratic ,linear or neither. For it be a quadratic relation the 2nd differences must always be constant, but if the 1st differences are constant then it is a linear relation. Furthermore, if both the 1st and 2nd differences are not the same that would mean that the equation is neither.
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In the example above it shows you that the equation is a quadratic relation because of its second differences being constant while the first differences aren't.

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:

  1. Labeling a parabola.
  2. 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.
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Transformation

When an equation is in vertex form each letter represents a 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)
Graphing Vertex Form Using Transformations

Findng the zeros in vertex form

Finding Zeros from Vertex Form

Vertex Form Word Problems

Economic Problem

Vertex Form word problem

Finding an equation given the vertex and two points

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Reflection

This unit was probably the one I like the most because I found it easy to understand and do. The whole concept of parabolas and transformations were very straightforward so I had no problem. Graphing had confused me in the beginning but I got the hang of it from look back at the lessons and teaching myself. Overall the whole unit was pretty fun and easy to do.

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


The function for factored form is y=a(x-r)(x-s)

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

A few things I picked up from the factored form unit was:

  1. Multiplying binomials by using foil.
  2. Factoring by grouping.

Zeros/X-intercepts/Roots

Finding the zeros in factored form is very easy, all you have to do is set y=0 and solve for x.

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

When multiplying binomials use "FOIL"

First term

Outside term

Inside term

Last 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

FOIL

Binomial Common Factoring

When you have 2 binomials that are exactly the same thing, add or subtract (depending on the signs) both the numbers and then multiply the 2 binomials that have now become one.


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

Factor Out a Common Binomial Factor

Monomial Commmon Factors

This method is essentially just factoring using GCF (great common factor). Look for common factors within the equation that can be divided evenly. After dividing remember to put the number and/or variable in front of the brackets to show that this is what you have divided the numbers within the brackets.



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 Common Monomial Factor

Factoring By Grouping

Factoring by grouping works when there are 4 terms that are able to be grouped by common factors.


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 by Grouping | MathHelp.com

Factoring Simple Trinomials

You can use this method to change an equation from standard form to factored form.

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)

Use decomposition when the a value in ax^2+bx+c does not equal to one. Multiply a by c to get the product and b is the sum, use product and sum to get the 2 numbers and then put them into the equation ax^2+(#1)+(#2)+c and solve by grouping.


For more help watch the video below

Factoring by Decomposition ( MyMathStudyGuides.com )
For more methods to factor complex trinomials, here is a video below on the guess and check method.
Factoring quadratic trinomials guess and check method

Special Cases

Factoring Trinomials - Special Cases

Graphing Factored form

Graphing factored form is easy! All you need to do is:

  • 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

This will be able to answer your questions or teach you about factored form word problems.
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Measurement Problems

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Reflection

This unit was the unit I find I had struggled the most with because of all the different ways to factor. When doing questions it wouldn't always tell you which way to factor so you had to figure it out which I had found kind of challenging. Practicing all the different types of factoring had helped a lot with this problem and had made factoring pretty fun be. Overall although at the beginning of the unit had been pretty confusing with practice and going through the lessons I had gotten better and understood a lot more of the concepts.

Standard Form

Completing the square

The quadratic formula

Discriminant

Graphing standard form

Word problems

Learning Goals

Even though this unit was easy and pretty fun, I learned many new concepts such as:

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

Completing The Square

Completing the square means to change an equation that is in standard form in to vertex form

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:

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Summary:


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

Equations in standard form can be solved using the quadratic formula. Using the quadratic formula can solve for the roots/zeros/x-intercepts. This is what the quadratic formula is:
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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.
Here is how to use the quadratic formula:
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Discriminant

The discriminate is a number that can be calculated from any quadratic equation that will tell you how many solutions there are. The discriminant can be calculated from the function b^2-4ac.


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.


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Graphing Standard Form

Graphing in standard form isn't difficult at all! All you you need to do is:


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

Word Problems

Measurement Problems

STANDARD FORM WORD PROBLEMS

Economic Problem

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Number Problems

Standard Form Number Problems

Reflection

This unit was very easy and also one of my favourite. Everything in this unit was straightforward and easy to understand. The quadratic formula was very easy because all you had to do was remember the equation and then plug in your a,b, and c value and completing the square was also an easy concept to get the hang of. At first we had started doing word problems which got me very confused because I didn't know what to do. I had struggled with the revenue problems the most because sometimes the wording of the problem would catch me off guard and I wouldn't know what to do or I would just make silly mistakes which would completely mess up my final answer. With practicing revenue questions and getting the help of my friends, I had gotten better at them and realized they were easy with much practice.