Quadratic Relations Website

Everything you need to know about Grade 10 Quadratics!

Made by Kamran Samra. Grade 10. Novemeber 2014

What is Quadratics anyway? Give an example

The word "Quad" means square, because the variable we use gets squared, like this:(x²).

For example, Quadratics can be used to graph the flight path of an object, for example, in football when you throw the football it makes a U shape and we can use quadratics to ensure that the ball will go in the direction and location you want it to go in. You can also use quadratics when it comes to business, you can use them in business meetings with charts when you want to make the highest amount of money possible and the highest point (the vertex) will be that mark. Quadratics are everywhere!

Table of Contents

Introduction to Quadratics

  • What is a parabola? (Basic Terminology)
  • Second Differences

Vertex Form y=a(x-h)²+k

  • Effects of Vertex form
  • Axis of symmetry
  • Optimal Value (y=k)
  • Transformations
  • Step Pattern
  • X-intercepts or zeros (Subbing in y=0 and solving)
  • Graphing Vertex Form

Factored Form y=a(x-r)(x-s)

  • Zeroes or x-intercepts (r and s)
  • Axis of symmetry x=(r+s)/2
  • Optimal Value (Subbing in)

Standard Form y=ax²+bx+c

  • Zeroes (Quadratic Formula)
  • Axis of Symmetry (-b/2a)
  • Optimal Value (Subbing in)
  • Completing the square to turn to vertex form
  • Factoring to turn to factored form


  • Common Factoring
  • Simple Trinomial
  • Complex Trinomial
  • Perfect Squares
  • Difference of Squares
  • Factoring by Grouping

Welcome to the world of Quadratics

What is a Parabola?

In the Quadratics unit, we call the graphed equations a parabola. All Parabolas have a:

Vertex- where the optimal value and the axis of symmetry meet. (The highest and lowest part of the line)

Axis of symmetry- the half line of the curve (vertical line that splits the parabola in half)

Optimal Value- the highest or lowest point of the parabola (determined by the vertex)

Y-intercept- the point when the curve crosses the y-axis ( all numbers above optimal value)

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. (Look at example on quiz below) If first differences are the same the relation is linear. If the second differences are equal are the same, the relation is quadratic.
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Now that we have a basic overview of Quadratics, Lets Begin with Vertex Form

Effects of Vertex Form. What is it?

When you have the vertex and the one of the x-intercepts of an equation. We can sub it into the equation y=a(x-h)²+k where the vertex is represented as (h,k) to solve for a.
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Axis of Symmetry

A line of symmetry for a graph. The two sides of a graph on either side of the axis of symmetry look like mirror images of each other.

Optimal Value

The Optimal Value in a parabola is the highest or lowest point in the graph. (The Vertex)

Transformations. Each variable represents a transformation

The (-h) value moves the vertex of the parabola left or right

When (-h) is negative move right and when its positive move left.

The (k) value moves the vertex of the parabola up or down

When (k) is negative move downwards and when its positive move upwards.

The (a) stretches the parabola and if the

If (a) is positive the parabola opens upwards, if negative it opens downwards.

Vertex= (h,k)

Step Pattern

Step pattern (a) value. determines the orientation and shape of the parabola (the stretch).
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Graphing Quadratics Using Step Patterns

x-Intercepts (y=0)

To find the x-intercepts in Vertex Form, we must sub y=0.

Example: y=4(x-3)²-1


0 = 4(x-3)² - 1
0 = 4(x² - 6x + 9)-1
0 = 4x² - 24x + 35
0 = (2x-5)(2x-7)
x = 5/2 and x = 7/2

x-intercepts (2.5,0) (3.5,0)

x intercepts from vertex form

Graphing Vertex Form y=a(x-h)²+k

Explanation shown in video, alongside example.
Step 1: Find the Vertex (h,k)

Step 2: Find the y-intercept (Let x=0 and solve for y)

Step 3: Find the x-intercepts (Let y=0 and solve for x)

Step 4: Graph the parabola using all 3 points (vertex, x-int, y-int)

Please see the video attached for clarification.

Graphing from Vertex Form

Factored Form y=a(x-r)(x-s)

Zeroes or x-intercepts (r and s)

A zero of a parabola is another name for the x- intercepts.

in order to find the x- intercepts you must set y=0

Example: y=-(x-2)(x+4) Each bracket represents a x-intercept.

0=(x-2) Move the -2 to the opposite side of the equal sign to isolate x and change it to +2

x=+2 (2,0)

0=(x+4) Move the +4 to the opposite side of the equal sign to isolate x and change it to -4

x=-4 (-4,0)

The zeroes (x-intercepts) are (2,0) (r) and (-4,0) (s) in the equation y=-(x-2)(x+4)

Axis of Symmetry x=(r+s)/2

To find the axis of symmetry, we must get our two zeros from before (2 and -4), add them and then divide them by 2. Using the following formula x=(r+s)/2

-4+2=-2 Add both the zeroes together.

-2/2=1 Divide the number by two.

Therefore the axis of symmetry is -1.

The image to the right shows what an Axis of Symmetry is (blue line)

Finding the Optimal Value (Vertex) using Substitution

To find the Optimal Value you have to sub in the axis of symmetry into the equation we had from the beginning. y=-(x-2)(x+4)




y=-(-3)(3) Expand the brackets with the a value (-1)


Vertex= (-1,9)

You can now get your Optimal Value and x-intercepts and plot them on the graph to create your parabola

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Please refer to the attached video for clarification on how to graph using Factored Form

3.5 Graphing from Factored Form

Standard Form y=ax²+bx+c

This quadratic formula is another method you can use to find the x-intercepts of an equation when it's in standard form.

Axis of Symmetry (-b/2a)

To find the Axis of Symmetry in standard form, we get have to get the -b value and then multiply the a value by 2, then divide. (-b/2a)

Example: 5x²+6x+1. b=6 a=5




Therefore the axis of symmetry (-0.6,0)

Zeroes (Quadratic Formula) x = -b ± √b² - 4ac / 2a

Example: 0=5x²+6x+1.

Identify the coefficients (a=5 b=6 c=1)

Now sub the values into the formula

x=-(6)± √(6)²-4(5)(1) / 2(5)

x=-6± √(36)-(20) /10

-6± √(16) /10

(-6±4) /10

-6+4/10=-0.2 (x-intercept (-0.2,0)


-6-4/10=-1 x-intercept (-1,0)

Two x-intercepts are are (-0.2,0) and (-1,0)

NOTE: You cannot take the square root of a negative number. (No x-intercepts)

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Discriminates (D)

A discriminate is the number inside of the square root in the equation ( -b ± √b² - 4ac / 2a)

It shows how many solutions (x-intercepts) the equation has


If the Discriminate is negative, there are no solutions to the equations

If the Discriminate is positive, there are 2 solutions to the equation

If the Discriminate is zero, there is 1 solution to the equation.

Completing the Square to Turn into Vertex Form

This method is used to rearrange the quadratic equation (ax²+bx+c) to a(x-h)²+k

Example: y=2x²+8x+5

First, Group like terms like (x² and x terms) together by putting brackets around them.


Second, Common factor (ONLY THE CONSTANT TERMS)


Third, Complete the square inside the bracket

y=2((x²+4x+4) -4) +5

(4/2)² ^ ^ Note: You need to subtract 4 so you don't make any changes to the equation!

Fourth, Write the trinomials a binomial squared . We must also expand into the brackets.

y=2 (x+2)² -8+5

Vertex Form: y=2(x+2)²-3 The Vertex (h,k) is -(2,-3)

Please use the video below for reference

Changing a Quadratic from Standard Form to Vertex Form


Common Factoring

Example: 8x+6

First, we have to find the GCF (Greatest Common Factor) which is 2

Divide 8 and 6 by 2.

Your new answer should be


Please refer to the example and video posted below

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3.11 Factoring

Simple Trinomials x²+bx+c

Here, we have the equation ax²+bx+c (when a=1) we have to find

2 numbers that add up to the b value

2 numbers that multiply together to give the c value

Example: x²-7x-18


So what 2 numbers that are added together equal -18 and added together equal -7



Therefore the answer is (x-9)(x+2)

For any further questions please refer to the video below

3.8 Factoring Simple Trinomials

Complex Trinomial x²+bx+c

Complex Trinomials have a coefficient other than 1 in front of the x² term.

There are two ways to factor Complex Trinomials, the first way is:


Example: 5x²+12x+4

The first and last value multiply together (5x4=20) We have to find what two numbers multiply together that add to 20 and add together to get 12 (10 and 2)

Now we replace 10x and 2x into 12x value and our trinomial now looks like


Now we have to find the GCF for the first and second brackets and divide them. First Pair=5x Second Pair=2


Therefore the answer is (5x+2) (x+2)

Trial and Error

Here, we have to guess and check and keep substituting numbers until we get the correct answer.

Example: 4x²+12x+9

First we start off with two brackets, we know that we can use (2x+__)(2x+__) that will get us 4x². Now we have to substitute numbers in that will get us 4x²+12x+9.

(2x+3)(2x+3) Expanded: 4x²+6x+6x+9 (Collect like terms and we get 4x²+12x+9)

For any further questions please refer to the video I prepared.

Factoring Complex Trinomials

Perfect Squares and Difference of Squares

Perfect of Squares include 1,4,9,16,25,36,49,64,81,100 etc.

First, the first number and third number can be square rooted

Example: 4x²+12x+9


√9 =3


Check your answer


Difference of squares

Two numbers in the expression where both numbers can be square rooted and the second number has to be negative.

Algebra - Factoring Differences of Squares

Sample Assesments

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This mini test was not written to the best of my ability, I noticed that I sort of understood most concepts but I just made a few silly mistakes that cost me to loose marks. I can fix these by practicing more and looking over my work.


In the quadratics unit, one of the biggest and most difficult unit we have done in the course thus far. I had learned a lot about the way they work and where they come in handy. For example, we see Parabolas in the cables that support bridges, architects will need to know this information in order to construct a bridge and make sure it is secure, we also see parabolas in roller coasters, they're literally everywhere! During this unit, i feel like i did average but i could have done better, at first i was VERY confused and then I had gotten help from my older cousin that started to help me out and studying with friends and it started to make a lot more sense. All in all, I had learned a lot and used to think to myself how stupid the unit was but now doing this website assignment and a bit more research about quadratics i learned how much I can benefit in life, learning about quadratics will help me become a better Engineer or Architect, my cousin is studying at university to go into the architecture field and she told me how important it is to know how quadratics work and how knowing it properly and understanding all the concepts will benefit me in the long run. I felt disappointed in myself at first because I did not do as well as i hoped to do, I felt as if i understood the concepts well, but there was just so many places to tie in and how each section was different and just got confused and didn't know how I could tie everything in place.

Word Problems

Now that we have covered all the topics in quadratics, we can start to use word problems where quadratics apply to the real world.

Motion Problems

Motion Problems focus on collecting data on the height and time of a curve, This can relate back to roller coasters, where the architects have to determine the amount of time it will take for a card to reach a certain point in the ride or when an athlete has to throw a ball to another player, (Kind of like the fireball activity). Please refer to the video attached that will show you how a motion problem can be solved.
3.12 Motion problems
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Area Problems

A rectangle has an Area of 204cm². The length of the rectangle is 5cm more than the width. Find the dimensions of the rectangle.
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Calculators are sold to students for 20 dollars each. 300 students are willing to buy them at that price. For every 5 dollar increase in price, there are 30 fewer students willing to buy the calculator. What selling price will produce the maximum revenue and what will the maximum revenue be?

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