Quadratic Relationships Index

By Ravdeep Gandhi

What is Quadratics:

Since elementary school you have only graphed linear systems which is straight lines, however now things are going to get a lot more complicated. In quadratics you do not graph straight lines but instead you graph curves which are known as parabolas. Throughout your interaction with my website you will understand how to create these curves using a variety of different equations being vertex form, factored form, and standard form. You will also learn how to use these equations to solve a multitude of questions.

Unit #1 Graphing Vertex form

Summary of the unit

In this unit you learn about the vertex form equation and how it tells you where the vertex of a parabola is located on a graph and how wide the parabola is. You also learn how to use the equation to graph a parabola and transition it in different directions. Finally you learn how to solve problems involving curved distances objects go through.

Learning Goals:

#1 Understanding the different parts of the Vertex form equation to be able to graph it

#2 Understanding the different parts of the Vertex form equation to be able to solve word problems with it

What the equation tells you:

Vertex Form is an equation that you can use to solve quadratic relations.

The equation being y=a(x-h) ²+k

The value a tells you the direction the parabola opens (positive=up, negative=down) and weather it is compressed (wider) or stretched (narrower)

The value h tells you the position of the vertex on the x axis (the opposite of h is the X-coordinate, for example if h is -1 then the x-coordinate is +1)

The value k tells you the position of the vertex on the y axis

What a graph with a parabola tells you:

Vertex: point where the axis of symmetry and the parabola meet

Optimal value: value of the y-coordinate of the vertex

Axis of symmetry: where the parabola divides into two equal half’s

Zeros: where the graph crosses the x axis (x-intercepts of the parabola). A graph can have no zeros if the parabola does not cross the x-axis.

Direction of opening: if the parabola opens up or down
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Linear and Quadratic Relations

First and second differences can tell you if the data in a table are part of a Quadratic relation or Linear relation

Subtract the previous y coordinate by the following y coordinate on a table until there are no y coordinates left to subtract.

If the differences (first differences) between the y coordinates are the same then the relation is linear. If they are different repeat the subtraction of y coordinates. If the new set of differences (second differences) are the same then the relation is quadratic. If both the first or second differences are not the same then the relation is neither linear nor quadratic.
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How to graph a vertex form equation:

vertex form

Graphing quadratics using step pattern:

Graphing Quadratics Using Step Patterns

Solving word problems:

word problems Vertex form Quadratic

Questions and answer from the video

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Graphing using transformations:

Quadratic transformations

Unit #2 Factored Form

Summary of the unit

In this unit you learn about the factored form equation and how it tells you what the x-intercepts of a parabola are and how wide the parabola is. You also learn many different ways of factoring which open you up to a lot of questions involving the area and perimeter of rectangles, along with previously mentioned questions of curved distances objects go through.

Learning Goals:

#1 Understanding the different parts of the factored form equation to be able to graph it

#2 Understanding and being able to factor in seven different ways

#3 Understanding the different parts of the factored form equation to be able to solve word problems

What the equation tells you:

Factored Form is an equation that tells you the x-intercepts of a quadratic relation.

The equation being y=a(x-r)(x-s)

The value a tells you the direction the parabola opens (positive=up, negative=down) and weather it is compressed (wider) or stretched (narrower)

The value of r and s give you the two x-intercepts (r,0) and (s,0)

How to graph a Factored form equation:

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Expand and Simplify:

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Monomial factoring (GCF):

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Binomial factoring (GCF):

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Factoring by Grouping:

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Simple Trinomial factoring:

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Complex Trinomial factoring

How to Factor (Decomposition)

Diffrences of Squares:

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Factoring perfect Squares:

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Application problem:

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Unit # 3 Standard Form

Summary of the unit

In this unit you learn about the standard form equation and how it tells you how wide a parabola is and what the y-intercept of the parabola is. You also learn how to graph the equation by using the quadratic formula to find x-intercepts of a parabola and learning how to complete the squares of a standard form equation to tell you what the vertex of the parabola is. You learn to solve word problems that involve consecutive integers, optimisation problems and expand on distance of curved objects problems.

Learning Goals

#1 Understanding the standard form equation to be able to convert it into a vertex form equation by completing the squares

#2 Understanding the standard form equation to be able to use it in the quadratic formula to find the X- Intercepts

#3 Understanding the standard form equation to be able to solve word problems with it

What the equation tells you:

The equation being y=ax²+bx+c

The value a tells you the direction the parabola opens (positive=up, negative=down) and weather it is compressed (wider) or stretched (narrower)

The value c tells you the y-intercept of the parabola

How to graph a standard form equation:

There are 3 parts to graphing a standard form equation

Part 1: Finding the X-intercepts using the quadratic formula

Part 2: Finding the vertex by completing the squares

Part 3: Graphing the y-intercept and mirroring that point on the parabolas axis of symmetry

Using the Quadratic equation to find the X- intercepts of a Parabola

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Where the Quadratic Formula came from

Deriving the Quadratic Formula

Completing the Squares to turn the Standard form equation into a Vertex form equation

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Discriminant

Discriminant expression: D = b² - 4ac

The a b & c vales from the standard form equation are subbed into the discriminate expression. After solving the expression the value of D tells us how many x- intercepts a quadratic relation has.

If the discriminate is positive, there are 2 x-intercepts

If the discriminate is 0, there is 1 x-intercept.

If the discriminate is negative, then there are no x-intercepts

Application problem:

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Reflection on the Quadratic unit

I didn’t really think much of the Quadratics unit because I knew in order to succeed in grade 10 academic math I just had to do it, and do it well. Quadratics was opening up an entire new realm of math concepts for me because for the first time ever I learned how to graph curves instead of straight lines. Learning the different forms a quadratic relation presents itself helped me grasp the concepts of quadratic relations better. Be they vertex form, factored form, or standard form they all give the product of a parabola.

In the first unit we learned the vertex form equation. This made sense because since this is the first unit, the best way to insure we succeed was to teach us the easiest form of a quadratic relation, vertex form. I found this the easiest unit from the three we were taught because all we had to do to graph a parabola was to read numbers from the vertex equation to place the vertex of a parabola on the graph and use the step pattern concept to create the width of a parabola. Also the word problems were not that hard because mainly all we needed to do was comprehend what the vertex equation told us about the quadratic relation and find the x/y-intercepts using substitution, a technique we learned last year.

The next unit we learned the factored form equation. This unit was the one I personally thought was the hardest because there were just so many concepts to learn. Starting off was easy because all we needed to do was expand and simply an expression. However when it got to the part that involved seven factoring It got a lot harder because we needed to learn to do so many different types of factoring (seven to be more specific). Also I thought that factoring was the most useless thing ever because all we were doing was making the expression more complicated. This was until I realised we needed to factor in order to graph the expression, and to solve a bunch of different word problems such as area and perimeter questions. Knowing this made me admire factoring a lot more but I still have a bleak hatred for it.

In the last quadratic unit we learned the standard form equation. This unit was also pretty hard because of the “completing the square” concept and word problems involving optimization and objects being fired from underwater. However completing the square was what made me understand that all these forms of factoring were connected. This is so because completing the square of a standard form equation turns it into a vertex from equation, which means equations can turn into other equations and give you the same parabola which I thought was pretty cool. Also the quadratic formula was a little difficult to understand because you needed to get your integers right, but after some fauilures with that I got a hang of it.

I’m not going to lie by saying this was my favourite unit in math, or what I learned in this unit was easy because in reality it was not. Quadratics has been the hardest unit for me in my entire math career, but that does not mean I haven’t mastered it. My proof of my mastery is from the factoring TIPS that involved me knowing all seven types of factoring well enough for me to create my own expressions and to check to see if those expressions were correct. Factoring was the hardest unit for me and the one I hated the most, so the fact that I got a level 4 on the TIPS that displays all the knowledge I learned from factoring shows how practising and asking questions can make you succeed in Quadratic units, and furthermore grade 10 academic math.