Quadratic Relations

By: Shayan Ahmed

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Table of Contents

What is Quadratics?

  • Introduction
  • Parts of a Parabola

Vertex Form

  • How to Graph

Factored Form

  • How to Graph
  • How to find the AOS
  • How to find the optimal value
  • Finding the x-intercepts

Using Polynomials

  • Multiplying Binomials
  • Standard Form ---> Factored Form

All Types of Factoring

  • Common Factoring
  • Factoring by Grouping
  • Simple Trinomial
  • Complex Trinomial
  1. Difference of squares
  2. Perfect squares

Standard Form

How to Complete the Square

  • Standard form ---> Vertex form

Quadratic Formula

  • Applying the quadratic formula
  • Discriminant


  • Revenue
  • Height & Distance
  • Area


Reflection and Assessment

External (Helpful) Links


Quadratic Relations can be graphically represented by a parabola, and they hold a Quadratic Relations can be graphically represented by a parabola, and they hold a constant second difference. This website will go through key features of the parabola, coordinates of the vertex, x and y-intercepts, and how to go from one form to another. When going through it, you will find links to websites, example problems, and lessons.constant second difference. This website will go through key features of the parabola, coordinates of the vertex, x and y-intercepts, and how to go from one form to another. When going through it, you will find links to websites, example problems, and lessons.

A quadratic relation has a second difference. If you have a linear relation (grade 9), you realize that is a first. The second difference of quadratic relations are the same. For a better understanding, refer to the diagram below

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Key Terms

In a quadratic relation, there as several key terms as follows

  • Vertex - The maximum or minimum point on the graph. It is the point where the graph changes direction

How to label ---> (x , y)

  • Minimum /Maximum Value - The highest or lowest point on the y- coordinate

How to label ---> y= ___ (min or max)

  • Axis of symmetry - A line which equally divides the parabola vertically

How to label ---> x= ___

  • y- intercept - Where the parabola intersects with the y- axis

How to label ---> y=___

  • x- intercept - Where the parabola intersects with the x- axis

How to label ---> x=___

  • Zeroes - same as the x-intercept

  • Opening - the direction in which the parabola opens. If the two arms are pointing up, it is opening up and vice versa.

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Vertex Form: y=a(x-h)^2=k

How to Graph:

The basic equation for a parabola is y=x^2. Although in vertex form there is a, h, and k. Let me show you what their jobs are.

  • Step Pattern: The basic step pattern is as follows, you take the value of y, and square it. So if y=1, x=1 or if y=3, x=9 etc.

  • a: Is the stretch factor in the equation. When "a" is changed, the step pattern of the equation changes. If "a" value changes, the step pattern is altered as well. If "a" is greater than 1, the line gets stretched. If "a" is less than 1, the line becomes compressed. Lastly, if "a" is a negative number, the graph is reflected on the x- axis.

  • h: Is the opposite reciprocal of the x-coordinate in the vertex. It moves the parabola left or right on the x-axis. If "h" is positive, it moves that many units to the left. If its negative, it moves that many units to the right.

  • k: Is the y-coordinate of the vertex. "k" has the ability to move the parabola up or down on the y-axis. For example, if the value of "k" is 6, it would move 6 up. If the value is -6 it would move 6 down.

Graphing a parabola in vertex form

Factored Form

How to graph:

Factored form is simple but it requires several steps to locate key

points which are then used to graph.

  • Direction of Opening: Is determined by "a", if "a" is positive, parabola will be opening up. If "a" is negative, it will open down.

  • Zeroes: In the equation, "r" and "s" are the zeroes/ x-intercepts. "r" and "s" are the opposite reciprocal of the zeroes. If the equation was, y= (x+4)(x+2), the zeroes would be -4, and -2.

  • Axis of symmetry: The axis of symmtery is the x-coordinate of the vertex, which brings you one step closer to completing your parabola. Using the given zeroes, the axis of symmetry is [(r+s)/2]. Using the example above, the axis of symmetry is -3.

  • Optimal Value: The optimal value is the y-coordinate of the vertex and the last step of factored form. In order to get the optimal value, you need to sub in the AOS. For example, y= (x+4)(x+2), the AOS is -3. Which makes it y= (-3+4)(-3+2), once solved, you get: min, y=-1.

Finding the x-intercepts:

The basic equation for factored form is y = ( x - n) ( x - n ) , where n is equal to any number. Much like the h value from the vertex form equation, n is always opposite in the factored form.

Look at the following, if an equation is y = ( x + 3 ) ( x - 5 ) , then the zeroes are located at (-3, 0) and (5, 0)

Graphing Parabolas in Factored Form y=a(x-r)(x-s)

All Kinds of Factoring

*Done by myself on word, transferred with a screenshot *
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Common Factoring

Common factoring

Simple Trinomial Factory

Simple trinomial factoring

Complex Trinomial Factoring

factoring complex trinomials

Using Polynomials

A polynomial is a "a polynomial is an expression consisting of variables (or indeterminates) and coefficients" - Dictionary. To make it simple, it's a expression which has multiple terms. The naming of polynomials comes down to the number of terms, eg. two: binomial. A common or possible binomial in quadratics can be, x^2+4x+20

Multiplying Binomials, two term expressions, is going from factored form to standard form, if you watch the following video it explains how it is done.

Multiplying Binomials (expanding)


Completing the Square

The following steps show how to go from standard to vertex form.

Using the picture below follow these steps:

  1. Is the standard form.
  2. Common factor out any term from the x^2 and the x.
  3. Using the "x" term, in this case "t", divide it by 2 and then square it. Since you are not allowed to change the equation, you have to add and subtract the number.
  4. Multiply out the subtracted value by the common factor and write it outside the bracket.
  5. Factor the perfect square (the number in the brackets) and rewrite it.
  6. The vertex form

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Applying the quadratic formula

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It is quite straightforward, all you do is take your standard equation and sub a,b,c into the quadratic formula. Once the roots/ zeroes are found, you can go back up the page where it shows you how to find the AOS and Optimal value, in order to get the vertex.

This is a good way to find the vertex and graph from standard form. If the discriminant is positive, you can easily graph.

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Revenue Problem

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Height&Distance Problem

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Area Problem

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All three forms

All the quadratic forms interact, they each are good for certain tasks. Factored form is a almost instantaneous way to get the zeroes. Vertex form quickly shows you the vertex and is easy to find. Standard form is good for quadratic equation and getting roots as well. They all can be graphed so that is one more common thing. The expression is a good way to draw a rough graph which helps to image a question before solving. To conclude, all of the forms are useful in their own way and are a advantage if used properly

Thinking Test (three pages)


I found that quadratic relations wasn't too much of a jump from grade nine, linear. I found it relatively easy to grasp onto topics once i tried them myself. Graphing basic parabolas using the vertex form was really easy, but when it got to factoring, all the types of factoring, equations such, it became more difficult. Those were three different concepts which all related, in the end at least. The types of factoring and all concepts were easy individually but when I was asked to apply it in certain questions I felt i didn't know what to use and when. Like for completing the square and quadratic formula, I had slight issues applying them. I would find zeroes but realize the question did not ask for that. When word problems were "worded" differently it confused me since it seemed like a whole new topic, but it was the same thing just trying to solve for something else. Also just making one calculation error throws off the whole question, which happened to me when factoring and expanding.

I did fairly well on some tests but I felt I could of done a lot better if I understood certain concepts better. The thinking test was probably my best test or one of the best, I was well prepared and made on minor, silly mistake. Otherwise I was satisfied.

Helpful Links