QUADRATICS

The Parabolic Adventure of Tanay Patel

Quadratics in life!

Have you ever wondered how a person in a human canon successfully completes the life threatening stunt?


The human cannonball is released in the air , reaches a maximum height, and then curves toward the ground and successfully lands on on his or her net.


But how?!?!?


The answer is quadratics!!!!!! In the ninth grade we learned all about straight lines. In Linear relations we learned how to graph a line using the formula "y=mx+b".


In the tenth grade, we are introduced to quadratic relations. This website talks about everything we learned in Quadratics. Feel free to leave any comments.

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Table of Contents

Introduction to Quadratics:

- Key features of Quadratic relations

- introduction to parabolas

- three ways to represent a Quadratic relation

-Types of equations: Vertex form

-Types of equations: Factored form

-Types of equations: Standard form


Vertex form:

- Investigating Vertex Form

- Graphing in Vertex Form

- Transformation of Parabolas

- Finding an Equation in Vertex Form using a Graph


MINI TEST #1


Expanding and Factoring

- Multiplying Binomials

- Common Factoring and Factoring by Grouping

- Factoring Simple Trinomials

- Factoring Complex Trinomials

- Difference of Square and Perfect Square Trinomials

- Finding an Equation in Factored Form Using the Graph

- Solving by Factoring


MINI TEST #2

Thinking Task- Factoring


Analysis of Quadratic Relations: Putting It All Together

- Completing the Square

- Discriminant and Quadratic Formula

- Evidence that the Quadratic Formula is Valid

- Word Problems- Flight


MINI TEST #3

- Multiple Choice

- Application

- Word Problems - Flight + Geometry

- Word Problems - Economics


Reflection


Useful Links

- Khan Academy

- Math is Fun

- Calculus Nipissingu

INTRODUCTIN TO QUADREATICS

Key features of a quadratic equation:

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Introduction to Parabola:

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  • A parabola can either open up or down.
  • Zeros can be either called "Roots" or "x-intercepts"
  • The zero(s) of a parabola is/are where it crosses the x-axis
  • The Axis of symmetry divides the parabola into two equal parts
  • The axis of symmetry (AOS) divides the parabola into two equal halve

  • The vertex of a parabola is the point where the AOS and the parabola meet.

  • The vertex is a y-co-ordinate which is optimal value

  • The y-intercept of a parabola is where the graph crosses the y-axis

  • The optimal value is the value of the y co-ordinate of the vertex

  • The vertex is the point where the parabola is at its maximum or minimum value.

Three ways to represent a quadratic relation:

Table Of Values

In the ninth grade we learned that if the first difference on the term value side of the table is constant, then the relation is linear. This year, we are taught a step further. When the first difference is not constant, then we look at the second difference. To find the second difference, we subtract the terms from the non-constant first difference.

Graphs

When you look a graph and the object on it is a curved line (parabola), the relation is quadratic.

The Power of 2

When a equation is given in the form of "ax^2+bx+c", the "^2" reveals that it is a quadratic relation. In a quadratic relation "ax" is squared.

Types of Equations

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Vertex Form

In the vertex form, an equation is given in the form of y=a(x-h)^2+k. As stated in the chart above, the variables h and k represent the x and y values of the vertex.

Investigating Vertex From

When the equation y=a(x-h)^2+k is given, an individual can explore a lot from it. Vertex form gives us the value of the axis of symmetry and the optimal value. The value of h gives a way the axis of symmetry. The value of k gives away the optimal value. Also, vertex form gives away the vertical/ horizontal stretch with help of the factor that represents a. If the value of a is greater than 1, the parabola is vertically stretched. If the value of a is less then 1, then the parabola is horizontally stretched.

Graphing in Vertex Form

Graphing vertex form
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Transformation of Parabolas

Transformations of parabolas
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Finding an Equation in Vertex Form Using the Graph

https://www.youtube.com/watch?v=yNprygJNHNU

Mini Test #1

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Expanding And Factoring

Multiplying Binomials

Multiplying Binomials
Let's take the example (x-3)(x+6)

When multiplying binomials always remember the acronym FOIL

First Terms

Outer Terms

Inner Terms

Last terms


Ok lets try this.


Step One: F

By multiplying the first terms, we are multiplying the two x's. And as everyone knows,

(x)(x)= x^2

Step Two: O

By multiplying the outer terms, we are simply multiplying x and 6.

(x)(6)= 6x

Step Three: I

By multiplying the inner terms, we are multiplying -3 and x.

(-3)(x)= -3x

Step Four: L

By multiplying the last terms we are basically multiplying -3 and 6.

(-3)(6)= -18

Step Five: Collect the like terms

We now have 4 terms: x^2, 6x, -3x, -18

What we do now is simply gather the like terms and add them together.

(x^2) + (6x) + (-3x) + (-18)= x^2 +3x-18

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Common Factoring/ Factoring By Grouping

Common Factoring anf Factoring by Grouping
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Factoring Simple Trinomials

x^2+5x+4


Step 1

Find 2 numbers that when multiplied have the product of 4 (the value of c) and when added have the sum of 5 (the value of b)


In our case, 2 numbers that have the sum of 5 and product of 4 are 1 and 4. 1 multiplied by 4 equals 4. 1 increased by 4 equals 5.


Step 2

Get rid of 5x (the middle monomial) and substitute it with 4 and 1 (the two numbers you found earlier that when added equal the value of b but when multiplied have the value of c)


x^2 + 4x + 1x + 4


Step 3

Factor. See the GCF between x^2 and 4x (first two terms) and 1x and 4 (second two terms)


x^2 + 4x + 1x + 4

x(x+4) + 1(x+4)

(x+4)(x+1)

Factoring Simple Trinomials

Factoring Complex Trinomials

Factoring Complex Trinomials

Difference of Square and Perfect Square Trinomials

Factoring Perfect Sqare Trinomials and Difference of Squares

Finding an Equation in Factored Form using the Graph

https://www.youtube.com/watch?v=fMtADYgVf-0

Solving by Factoring

OK!! So when we learned factoring, we learned that the final solution comes in 1 or 2 brackets. For example: y=(x+3)(x6) , y=(2x+3)^2 , y=(2x+5)(2x-5) , etc. But what do we do with this information? Well we know that the equation is in factored form. So maybe knowledge about factored form will come to use.


We know that the values of r and s in factored form represent the zeros of a parabola. in the same way, when solving by factoring, we are trying to find the zeros of the parabola.


For instance, Let's take the equation y=(x+3)(x-5)

To solve this equation, we have to replace the variable of y as 0.

Therefore the equation will be: 0=(x+3)(x-5)


Therefore x+3=0 and x-5=0

And when x is ISOLATED, x=-3 and x=5. So with this information we now know that the two x intercepts are -3 and 5.

MINI TEST #2

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Thinking Task - Factoring

Analysis of Quadratic Relations: Putting It All Together

Completing the Square

https://www.youtube.com/watch?v=_v94KclNHY4

Discriminant and Quadratic Formula

The Discriminant and Quadratic Formula are probably the most important subtopic to know when learning Quadratics. The Quadratic Formula is used to find the zeros of a parabola. The Discriminant as used to analysis whether we have two zeros, one zero, or none.
Discriminant and Quadratic Formula

Evidence that the Quadratic Formula is Valid

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Word Problems- Flight

Word Problems - Flight

MINI TEST #3

Multiple Choice

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Application

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Word Problems - Flight + Geometry

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Word Problems - Economics

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Reflection

Overall, I found that quadratics was an interesting unit. Exploring parabolas and connecting basic quadratic fundamentals with everyday life was not only a learning experience but a life changing journey.


Many teenagers have that one question in their head during math class: "When will this every help me in life"? Well... the truth is that it might not... BUT learning something as hard as quadratics with out a teacher would be next to impossible for the average student. Taken a psychological point of view, things that matter such as raising a child or doing taxes can be learned by an individual on their own, but learning functions would be difficult for an individual to learn on their own. Our generation spends more time in the local McDonalds than they would spend studying math at home. Yet many of them don't know that the logo of the poisonous fast food restaurant is made up of two PARABOLAS. Now I'm not saying that it is important to know junk like that. Although, not only would it be nice to know where the food comes from but also where the hypnotic logo (that intimidates millions of people every month) is traced back to.


Anyway back on track, math is every where, not only in a math text book. The bridge that you crossed over this morning, had a lot to do with functions in general, the flight most of your parents had to take to immigrate to Canada, had to do a lot with functions and physics, and the maximum revenue that your mom or dad would make at their store, has a lot to do with application of quadratic fundamentals.


To end off, I would like to say that even though quadratics was one of the most basic functions in math, I look forward to using my fundamental knowledge in future math courses.

Useful Links