Quadratic Relationships

By: Saarukha Rajhkumar

Learning Goals

What Can I Accomplish?

  • I am able to graph quadratic functions giving critical information and transformations where appropriate.

  • I am able to gather information from the vertex form of a quadratic (direction of opening, axis of symmetry, vertex, step pattern)

  • By given a set of transformations from the parent parabola, I am able to graph and state the equation of the parabola in vertex form.

What Is Quadratics?

Quadratics is all about parabolas. Parabolas are curves, meaning at one point -called vertex- the line turns. We use three different quadratic formulas to create, form and analyse these curves.

  • Vertex Form=a(x-h)^2+k
  • Factored Form=a(x-r)(x-s)
  • Standard Form=ax^2+bx+c

This flyer will help you better understand the quadratic equations, with lessons, videos and examples.

Down below, these are some of real life examples of parabolas we see in our lives

All About Quadratic Relations

Relations that are non-linear are known to be quadratic

A quadratic relation is :

  • A relation that has an equation (y=ax^2+bx+c)
  • Where a, b and c are real numbers
  • Where a ≠ 0

While linear equations are a straight line, a quadratic relation is a curve. Here are some new vocabulary words for this unit and a labelled diagram of a parabola.

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Vertex Form

4.1- Investigate Non-Linear Relations

What are linear relationships?

A linear relationship is a trend in the data that can be modelled by a straight line that shows a steady rate of increase or decrease.

What are non-linear relationships?

A non-linear relationship (quadratic) is a trend in the data that can be modelled by a curve. This curve is called a parabola.

Below, is an example of a linear and a non-linear relationship

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Lesson 4.2 -Quadratic Relations

First and Second Differences

In order to determine if an equation is linear or quadratic we must look at the first and second differences. First differences are the differences in the y-values chart.

How to Calculate First and Second Differences

In order to calculate the first differences in the y-values chart you should use the equation (y2-y1) where you subtract the top number from the bottom number. If the first differences are constant/same that means the relation is linear. Suppose the first differences aren't the same/constant you have to take the same steps as the first differences but this time you're subtracting the new numbers gotten from calculating the first differences rather than the existing y-values. If the second differences are same/constant that means the relation is quadratic. In some cases if you do the first differences and the second differences and you do not get constant/same numbers that means the relation is neither linear or quadratic.

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Linear, Quadratic or Neither using First and Second Differences Tables

Lesson 4.3- Write the equation in vertex form given transformations

What Is Vertex Form?

The transformed function y=a(x-h)² +k is the vertex form of the quadratic equation. The axis of symmetry (AOS) is represented by the h, and the vertex is represented by (h,k).

Transformations of Vertex Form:

  • The "a" value= Determines the stretch of the parabola, however if the "a" value is a fraction or a decimal it will compress the parabola. If the a value is a whole number then the parabola will get stretched. Lastly, if the a value has a negative sign in front of it then it will be reflected in the x-axis.
  • The "h" value= Determines the horizontal translation to the left or the right of the parabola. If the number is negative it moves to the right side (positive) and if the number is positive it moves to the left side (negative)
  • The "k" value=Determines the vertical translation, meaning how many units the parabola will move up or down

Here is an example:

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Equation: y= 15 (x+5)²+ 7

  • opens up (minimum at 7)
  • vertically stretched by a factor of 15
  • translated 5 units down
  • translated 7 units right
  • vertex: (-5,7)

Graphing Quadratic Functions Using Transformations

Lesson 4.4 -Graphing in Vertex Form

Mapping Notation

x | y= x² ← This is the notation for the base graph, y=x²

-2| y= (-2)²= 4 Using this mapping notation, you can plot the points onto the graph,

-1| y= (-1)²= 1 connect the lines and create a parabola.

0 | y= (0)²= 0

1 | y= (1)²= 1

2 | y= (2)²= 4

Mapping Formula

By using the mapping formula, you can go from the graph of y=x² to y= a (x-h)²+ k...

(x , y) → ( x+h, ay+k )

Example: y= (x-4)² +2 [a=1, h=4, k=2] (x,y) → (x+4, y+2)

Steps to Graph

1.Write mapping formula (x,y) → (x+h, ay+k)

2. Complete table of values for y=x²

3. Determine transformed "key" points (x+h)

4. Sketch base function, then new graph

Example: y= 2(x-3)² +1 [a=2, h=3, k=1]

(x,y) → (x+3, 2y+1)

x | y= x² → x | y= 2(x-3)²+ 1

-2 | 4 > (-2+3) > 1 | 2(4)+1 = 9

-1 | 1 > (-1+3) > 2 | 2 (1)+1 = 3

0 | 0 > (0+3) > 3 | 2 (0)+1 = 1

1 | 1 > (1+3) > 4 | 2 (1)+1 = 3

2 | 4 > (2+3) > 5 | 2 (4)+1 = 9

Step Pattern

When using step pattern to graph the parabola, multiply (a) by 1,3,5, etc.
The product you get is the rise and the run increases by one each step.

Example:Sketch a graph for y= 3 (x-2)²- 9

  • Vertex: (2,-9)
  • a=3
  • Step1 1x 3 = 3 (3/1)
  • Step2 3x 3 = 9 (9/2)
  • Step3 5x 3 = 15 (15/3)
  • A.O.S: x = 2
  • y-intercepts: x=0

y= 3(0-2)²-9

= 3(-2)²-9

= 3(4) -9

= 12-9

= 3


Graphing Quadratics Using Step Patterns
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Graphing a parabola in vertex form | Quadratic equations | Algebra I | Khan Academy

Finding Zeros and Y-intercepts

To find the zeros (x-intercepts) and y-intercepts, all you need to do is substitute 0 into x or y. While finding the x-intercept, remember that when you square root the term on the left, there can be two possible square roots (1 positive and 1 negative) meaning there will be 2 x-intercepts.


y= 2(x-5)²- 50

Y- int (x=0) | x- int (y=0)

y= 2(0-5)²-50 | 0= 2(x-5)² -50

= 2(-5)²- 50 | 50= 2(x-5)²

= 2(25)-50 | 2 = 2

= 50-50 |√25= √(x-5)²

= 0 |5+5=x -5+5=x

(0,0) | 10=x 0=x

(10,0) (0,0)

Finding x-intercepts (Vertex Form)

Word Problem

Quadratic word problems typically ask for this information:

Objects in the Air

• What was the initial height?

Find the y- intercept (x=0) and solve for y (the height when time=0)

• How long was it in the air?

Find the x- intercept (y=0) and solve for x (the time when ball hit ground, height=0)

• When did it hit the ground?

The time when ball hit the ground (y=0) so the x-intercept

• What is the maximum height of it?

The y- value of the vertex is the maximum height (y=k)

• At what time did it reach the maximum height?

The x- value of the vertex is the time when it reached maximum height (x=h)

• Height of it at 3 seconds?

Substitute 3 into the x-value(t) of the equation and solve for y(h)


At a baseball game, a fan throws a baseball back onto the field. This is modelled by the equation h=-5(t-2)² +45. (h= height in metres, t= time in seconds).

a) What was the initial height? (t=0)

h=-5(0-2)² +45




∴ The initial height was 25m.

b) What was the height after 3 seconds?





∴ The height of the ball was 40m after 3 seconds.

c) What was the maximum height?

The maximum point is the y-value of the vertex, (2,45), so the max height is 45m.

d) What time did the ball reach its maximum height?

The time the ball reached its maximum height is the x-value of the vertex, (2,45), so the time when the ball reached its maximum height is at 2 seconds.