Quadratics relations

by Amandeep Pabla, grade 10 (period 2)

What are quadratic relations? how we use it in real life?

The name Quadratic comes from "quad" meaning square, because the variable gets squared (like x2).It is also called an "Equation of degree 2" (because of the "2" on the x).Some real life examples can be: Mcdonald's logo , rides at wonderland, bottom of Eiffel tower, a football throw up in the air, and many more. Architects uses this unit of the grade 10 math to build these beautiful and amazing buildings, towers and more in this world.

Some examples of real life parabola

Table of contents

  1. Basic information about quadratics
  • Analyzing Quadratics
  • Solving Parabola's
  • Transformations of parabola



2. Different types of equations

  • Vertex form
  • Factored form
  • Standard form



3. Vertex Form

  • Graphing from vertex form
  • Finding equations of parabola given the vertex
  • isolating x

4. Factored Form
  • Factoring
  • Common Factoring
  • Factor quadratic expressions of the form
  • Factoring simple Trinomial
  • Factoring complex trinomial
  • Factoring by grouping
  • Special factoring cases

-Differences square

-Perfect square

  • Solving quadratic equations by factoring
  • Graphing quadratics in factored form
  • Factored form of the quadratic relation

5. Standard form

  • Complete the square (standard to vertex)
  • Solving quadratic equations from standard form
  • Discriminant


6. Word problems


  • Motion word problem
  • Geometrical word problem
  • Economical word problem
  • Number word problem

Introduction

Analyzing quadratics

Starting with table of values, table of values decides whether the equation is a quadratic relation or not. Check what table of values is in the picture and the video below. In quadratic relation 1 difference will always be different and it will decrease with a same amount of number each time. For example: -3,-1, 1, 3. this also relates to grade 10’s first unit called linear systems. In that unit we learned about linear lines. It relates to quadratic relation because the sometimes the first relation are same therefore it a a linear line.

Second differences

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3.1 Analyzing Quadratics

Solving Parabola's

Key features of quadratics:


  • Vertex (intersection of A.O.S. and parabola)
  • Axis of symmetry (x-axis)
  • y-intercept (where the parabola meets the y axis)
  • Zeros ( x-intercepts)
  • Optimal value (y value of the parabola)



Things to know about parabola:

  • Parabola can open up or down.
  • The zero of a parabola is where the graph crosses the x-axis, "Zeroes" can also be called "x-intercepts" or "roots".
  • The axis of symmetry divides the parabola into two equal halves.
  • The vertex of a parabola meet. It is the point where the parabola is at it's maximum or minimum value.
  • The optimal value is the value of the y co-ordinate of the vertex, the y-intercept of a parabola is where the graph crosses the y-axis.
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Transformations of a parabola

Some appropriate terminology of transformations are:


  • Opens up/down (reflected in x/y axis)
  • Vertical stretch by factor of a number
  • Horizontal translation of units right or left
  • Vertical translation of units up or down

Vertex form

y=a(x-h)²+k

Graphing from vertex form

Key points:

y=a(x-h)²+k

a- Stretch of parabola

h-moves the vertex left or right

k-this moves the vertex up or down


Watch the videos below to know how to graph an equation in vertex form on a graph paper and on Desmos.

5.1 Graphing Quadratic Equations in Vertex Form

Finding equations of parabola's given the vertex

Vertex=(h,k)

y=a(x-h)²+k


Example 1:

Vertex @ (2,4)

y=(x-2)²+4 \\ h and k value has been subbed into the equations


Example 2: (also finding the a value)

Vertex @ (-2,8) x-intercept (2,0)

y=a(x-h)²+k \\ basic formula

y=a(x+2)²+8 \\subbing the vertex

0=a(2+2)²+8 \\subbing the x-intercepts

0=a(4)²+8 \\ added the 2+2 inside the bracket

0=a16+8 \\ squared the 4 inside the bracket

0-8=16a+8-8 \\ moved the 8 on the other side

-8=16a \\ 8 became -8

-8/16=16a/16 \\ divide both sides by 16

-1/2=a \\ found the a value and now put it into an equation


therefore: y=-1/2(x+2)²+8


Also watch the video to know how to find equations when parabola is given.

3.4 Finding the Equation given Vertex

Isolating x

In isolating x your suppose to find what the value of x is. There are always to answers for this problem
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You can also watch the video!

solving quadratics from vertex form

Factoring

Factoring is a process by which a the factors of a composite number or a composite expression are determined, and the number or expression is written as a product of these factors.Factoring is an important process in algebra which is used to simplify expressions, simplify fractions, and solve equations.
IMG 6715 MOV

Common factor

Common factoring is extremely important in this unit because every question in this unit you will need to ask yourself "CAN I COMMON FACTOR?" in order to get the correct answer. Common factor is basically factoring something common inside the bracket. Take a look at the examples and the video.
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Common Factors

Factoring simple trinomial

In simple trinomial we will factor trinomials that have a lead coefficient of 1. Take a look at the examples and video below:
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Simple trinomial

Complex trinomial

Complex trinomial is similar to simple trinomial but in complex trinomial you have a trinomials whose lead coefficient is greater than 1. Take a look at the examples and a video.
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Complex trinomial
3.9 Complex Trinomial Factoring

Factor by grouping

In factoring by grouping, you will need to look at 2 key points:


  1. The equation has 4 terms
  2. Arrange the middle terms so you can factor.


Take a look at the example below.

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Special factoring cases

  • Differences square
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  • Complete square

It is the opposite of differences of squares

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Solving quadratic equations by factoring

Solving quadratic equation is finding the x-intercepts of the equations.
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Solving Quadratic Equations by Factoring.avi

Graphing quadratic in factored form

In order to put the quadratic relation in factored form you need to factor. Take a look at the example and video.
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Solving Quadratic Equations by Factoring - MathHelp.com
  • Now graph this by using Desmos on your phone or on your computer.

Complete the square

Completing the square is basically telling us how to go from standard form to vertex form.

Take a look at the examples and video.

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3.14 Completing the square

Solving Quadratic equations from standard form

This formula is very helpful for solving quadratic equations that are difficult or impossible to factor, and using it can be faster than completing the square. The Quadratic Formula can be used to solve any quadratic equation of the form ax2 + bx + c = 0.

The formula:
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How to Solve Quadratic Equations
How to Solve Quadratic Equations (pt.2)
  • If you think you can't remember the formula? Take a look at the video below
How to Remember the Quadratic Formula

Discriminant

The number D=b²-4ac determined from coefficients of the equation ax²+bx+c=0. The discriminant revels what type of root the equation has.

Note: b²-4ac comes from the quadratic formula


Take a look at the video to know how to solve discriminant problems

The Quadratic Formula: How to Use the Discriminant
The Quadratic Formula: How to Use the Discriminant to Determine Roots

How to find the vertex in standard form when there is no x-intercepts

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Word problems

Why do we need to know word problems?


  • Word problems teach you to accept a challenge but also to persevere and use both logic and creative ability combined. When you visualize the word problem you will be able to get to the solution quicker. Most of us enjoy a good challenge; after all, it gives spice to life doesn't it. It makes a boring day interesting and a good challenge can even revitalize all your senses.

Take a look at the video below to know how to solve any kind of word problem.
easy system to solve word problems.wmv

Now take a look at some word problems in quadratics

1.Motion word problems

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Take a look at a another example in the video below
Application Problem with Quadratic Formula

2. Geometrical word problems

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Solving a Geometry Word Problem by Using Quadratic Equations - Example 1

3.Economical word problems

3.13 Quadratics Economics

4.Number word problems

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Connection between the topics

  • Simple trinomial and complex trinomial

The connection between simple trinomial and complex trinomial is that there is a number front of x so that makes it ax²+bx+c where simple trinomial is x²+bx+c or else the method of solving simple and complex trinomial is same.

  • Zeros and discriminant

Finding the zeros is when you get two x- intercepts and in discriminant you always will either have no x-intercepts, 1 x-intercept or 2 x-intercepts. The concept of x-intercept is same, therefore the idea of x-intercepts connect these topics.

  • Standard form, Factored form and vertex form

All these topics are connected because all these equations make parabolas, solve parabolas and graph parabola. We can also convert these equations into each other therefore these 3 topics are connected.

Summary of the unit and how I did in this unit

Here ends this unit of quadratics, I learned a lot of new things from this unit, this was one of the challenging unit in grade 10 math so far. In grade 9 I didn’t even know what parabola were and never noticed that parabolas are everywhere. I started this unit by just analyzing quadratics. This just told me what parabola is and how it works. Then I learned about second differences, in grade 9 I only learned first differences. Then I learned how to solve parabola and the key points like Vertex, Axis of symmetry, Y-intercept, Zeros and the optimal value and where they located. I started this of well but I had some difficulties in transformations and how to actually right proper terminology of transformations. Then I moved on to Vertex form. Starting off with graphing from vertex form. I didn’t have any difficulties doing the graphing because I understood the key points of parabola really well, therefore that made graphing easy. Then I learned how to find equations of parabola given the vertex and isolating X, I didn’t have really have any problems understanding it because I kept on doing my homework on time and I was always paying attention in class. Then moved on to factoring. I had a lot of problems in staring of this unit. Starting off with common factoring, I had a lot of difficulties understanding the concept of common factoring. The class moved on to simple trinomials, complex trinomials and factoring by grouping but I didn’t really understand the concept of common factoring but then I went for extra help from my math teacher at lunch. That way I understood the concept and caught up in class. Then I learned special factoring cases. In that differences of square and completing square. I didn’t have any problems understanding the concept. Then moved on to solving and graphing quadratic equations by factoring form, I didn’t have any problems. Then I learned about complete the square. I had a little problem understanding these kind of problem but my tutor helped me with this so I understood these kind of problems. Then moving on to standard form, In standard form I learned solving quadratic equations from standard form, Discriminant and how to find the vertex in standard form when there is no x-intercepts. I didn’t have any problems understanding these concepts. Overall, this unit was really challenging and fun. I really understood this unit very well, I hope this unit helps me in life.

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