Quadratic Relations!

With just a few scrolls, learn all about Quadratics!

The Importance of Quadratics

Ever wondered where a quadratic relation would appear in your everyday life? Well, take a look around and scan your surroundings. The cables that support suspensions bridges, roller coasters, and the path that a foot ball takes when thrown into the end zone are all parabolic. Quadratic relations and parabolas can be very helpful if for example an architect wants to determine the curve of a cable in order to support a bridge. There are many uses for quadratic relationships in the business field, which is one of the main reasons that it is apart of our curriculum,

Quadratics Guide

Intro to Quadratics

  • Analyzing properties of a Parabola
  • Second Difference
  • Step Pattern

Factored Form

  • How to use Algebra Tiles (A quadratic relations rookie's life saver)
  • Expanding Binomials
  • Common Factoring
  • Factoring Simple Trinomials
  • Factoring Complex Trinomials
  • Special Cases (Perfect Squares, Difference of Squares)
  • Factoring by Grouping
  • Graphing Factored Form

Vertex Form

  • Effects of Vertex form
  • Isolating X
  • Determining (a) value (given a point and vertex)
  • Graphing Vertex Form

Standard Form

  • Completing the Square
  • Solving with Quadratic Equation
  • Discriminant
  • Finding the Axis of Symmetry (x value of the vertex)
  • Graphing Standard Form


Word Problems


  • Motion Problems
  • Area Problems


Reflection

(My thoughts on the unit)

Lets Get Started!

Anaylzing the properties of a Parabola

In Quadratics, we call the graphed equations a parabola. Parabolas have a number of properties that are present when they have been graphed. These include:


Vertex- where the optimal value and the axis of symmetry meet

Axis of symmetry- the half line of the curve (vertical line)

Optimal Value- the highest or lowest point of the parabola (horizontal line)

y-intercept- the point when the curve crosses the y-axis ( all numbers above optimal value y< and all numbers below optimal value y>

x-intercepts/roots/zeros- the points where the curves cross the x-axis (all real numbers)

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Second Differences

In order to determine whether table of values is a quadratic relation or not, you can simply take at the second differences of the the numbers in the table. Quadratic equations always have an equal second difference, however, always have an unequal second difference. Take a look at the picture shown on the right.
Step Pattern


The step pattern is applied in all graphs. The step pattern helps determine where each of the points of the graph are going to be. The points on the graph will always be affected by the (a) value in your equation. If there is no a value, then you are to use to original pattern, however, if there is an a value, that value will be multiplied to each value in the original step pattern chart. Refer to the video.

Graphing Quadratics Using Step Patterns

Factored Form

An equation written in Factored Form is written as y=a(x+r)(x+s)

The (r) and the (s), represent the x-intercepts when graphed.

The vertex is written as (x,y) meaning (Axis of symmetry, y-intercept), once the values have been found

The (a) value is multiplied to the step pattern, to determine the coordinates on the graph



Algebra Tiles

For beginners and visual learners, algebra tiles are a great and effective way that can solve for quadratic relations. Refer to video.

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Expanding Binomials

Learning to Expand binomials, basically going from factored form to standard form is very important. this transition makes it much easier to graph the equation in vertex form. Refer to the video in order to see the step by step process of expanding binomials. Refer to the video to learn how expanding binomials is done.

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Common Factoring

There are many different ways that you can factor an equation. For learning about any method of factoring, it is important that you know how to common factor. Refer to the video.
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Click on this very help full link from Khan Academy in order to work on some practice questions!
Simple Trinomials

(Possibly the easiest of the bunch)

Simple Trinomials
Complex Trinomials

Complex Trinomials uses the guess and check chart. The first set of numbers that multiply to your first number in your expression are cross multiplied with whatever two other numbers that can be multiplied to equal your third number in your expression. Then, in order to determine whether the pairs equal the final expression, simply add the two numbers in the last column of the guess and check chart. If the two added numbers equal the second number in your expression, you have found your answer. Refer to the video

Complex Trinomials
Perfect Squares and Difference of Squares

Perfect Square- Three numbers in the expression, First number and third number can be square rooted. In order to prove if the expression is a perfect square, multiply your square rooted numbers results together, then multiply them by 2. If the answer ends up adding to the second number in your original expression, you have solved your perfect square correctly.


Ex. 4x2+12x+9

4x2 square rooted= 2x

9 square rooted= 3

(2x+3)2

Check

(2x)(3)(2)= 12x


Difference of squares- Two numbers in the expression. Both numbers can be square rooted, however, second number is negative.

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Factor by Grouping

Note that factor by grouping can only be done when there are four numbers in the expression, making it very easy to determine how you will be solving the expression.

Factor by Grouping
Graphing Factored Form


When working with Factored form, the equations will always look as such: y=a(x+r)(x+s)

If the equation is not given in factored form, determine what type of factoring (simple trinomial, complex trinomial, difference of squares e.t.c) must be done to the equation, and bring the equation to Factored form y=a(x+r)(x+s)

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* Keep in mind, the step pattern does not play a huge role in Factored Form, as the main goal for Factored Form is to determine the vertex and the x-intercepts. However the step pattern can be used for the curves accuracy. The step pattern plays a larger role when working with the next type of relation; Vertex Form
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In order to plot an accurate graph, use the very helpful website/app!
Factored Form leads us into Vertex Form, another huge part of quadratics. Vertex form shares some of the same methods of isolating certain variables in order to determine values.

Vertex Form

An equation in Vertex Form is written as such; y=a(x-h)2+k

The (h) and (k) value represent the Vertex. It is written as (h,k) representing (Axis of symmetry, y-intercept).

The (a) value is multiplied to the step pattern in order to determine the coordinates on the graph.

The x-intercepts are represented by (x,y)


Effects of Vertex Form (Properties of Vertex Form)


The (-h) moves the vertex of the parabola left or right. Keep in mind that if the (h) is negative, it is to be moved right, and when the (h) is positive, it is to be moved left.

This horizontal shift, affects the axis of symmetry.


The (k) moves the vertex up or down. Keep in mind that if the (k) is positive, it is to be moved up, and if the (k) is negative, it is to be moved down. This vertical shift affects the optimal value.


The (a) is a vertical stretch or compression. if the (a) is positive, the parabola will open up, if the (a) is negative, the parabola will open down.


The vertex in vertex form is made up of h and k and is written as (h,k).

Isolating X

Isolating x is used in vertex form when trying to find the x-intercepts of the equation. The steps below show the steps that need to be taken in order to find the zero's (x-intercepts).



Note that the original equation for every equation in vertex form is written as

y=a(x-h)2+k

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Determining the a value (Given a point and the vertex)


In vertex form the a value highly impacts where the points of the graph will be as the a value is multiplied to the step pattern. However in certain equations, the (a) value, will not always be given. In these circumstances, the equation will always have a point (x,y) and the vertex (h,k). Down below there is an example of how you can isolate the (a), in order to find its value.

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Graphing Vertex Form

Graphing Vertex Form is the easiest out of the three. In the different subsections above, you can see the different situations the an equation of vertex form can be in. Always keep in mind the the equation of Vertex form is y=a(x-h)2+k. Here is a video, that will show you how to graph Vertex form.


3.2 Graphing from Vertex Form
Click on the link to complete a few questions on graphing vertex form!
Vertex Form is one of the easiest ways to graph a quadratic equation. Although there are three types of equations, standard form can easily be converted into vertex form.

Standard Form

A Standard form equation is written as y=ax2+bx+c

or

A standard form equation can also be written as ax2+bx+c=0 < meaning that the y has been turned into a zero


Completing the Square


Completing the square is a quick and easy way to transform a Standard form equation into a Vertex Form equation. This is helpful as graphing in Vertex form is known to be the easiest of the three ways. In order to complete a square, you must know how to factor perfect squares. Refer back to the Perfect squares video in the Factored form section to refresh your memory on how to complete a square.

*Keep in mind, a Perfect square consists of three numbers (ax2+bx+c) where the first and second number can be square rooted, and the results of the square rooted numbers multiplied together, then also multiplied with 2 are equal to the second number in the equation.

Watch the video below to learn how to complete a square.

3.14 Completing the square
In order to get a better understanding on how to complete a square, click on the helpful link below!
Along with converting Standard Form to Vertex Form, Standard Form can also be easily plotted onto a graph using the Quadratic Equation and isolating certain variables to find the vertex
Quadratic Equation

Quadratic Equation is used on Standard form equations, if the equation is not easily factor-able, meaning that a number that needs to be factored, would result with a decimal.

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For more information about The Quadratic Formula, visit the link below
Discriminant


Discriminant is apart of the Quadratic Equation of which it allows us to determine how many solutions (x-intercepts) a curve will have, when plotted on a graph. This is found within the quadratic formula

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Finding the Axis of Symmetry (x value of vertex)

In order to find the x value of the vertex use the formula -b/2(a)


Example: a=3 b=3 c=3 x= -3/2(3)

x= -3/6

x= -0.5 --------> (-0.5, 0) is the x value of the vertex


Graphing Standard Form
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Word Problems

Now that you have been introduced to all things quadratics, wrapping the whole unit together with word problems is a great way to tie all the pieces together.


Motion Problems


Motion Problems Focus on collecting data on the height and time of a curve. This can be represented by a roller coaster in our everyday life, if you were to calculate the amount of time it took a cart to go up a ramp, and come back down. This video will show you how a motion problem can be solved.

3.12 Motion problems
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Area Problems
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Reflection

This quadratics unit has been one of the biggest units that I have ever been taught. Just as expected, a lot of information was gained throughout these last months. I believe the equations and techniques we learned will not only better us as math students, but can also highly benefit those interested in going into the architect world and engineering. I believe out of all units that I have been taught, Quadratic equations will be one that I remember. This unit was defiantly challenging for me. I felt as if I understood all the concepts, however had a hard time putting them together. There are so many different ways that the different topics in this unit somehow were all in the end connected. This unit was linked and allows you to convert a majority of questions into the formula you prefer in order to plot it on a graph or even to over view information. In this unit, I believe I struggled most with communication. I also struggled with application as well. I find that it is very hard for me to complete word problems because it was hard for separate the information given. There are also many different types of ways to solve different equations, which did confuse me at some points because I was not sure what method was suppose to be used. There were also many formulas in need of memorizing. However, creating this website has allowed me to see the different connections between the different parts of the unit, and also helped get a grasp on certain methods that I was confused about before. I enjoyed doing the different types of factoring the most because in the end. in order to graph factored form, you ultimately needed to have two terms. In terms of graphing, I enjoyed Vertex form the most. The equations of vertex form had most of the information needed to plot the equations. If information was missing, small steps would be needed to be taken, in order to plot the graph. I also found the discriminant was very helpful, as it allowed me to determine whether the extra steps of finding the roots would be needed to be taken, by determining the amount of solutions present.
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