Quadratic Relationships

By: Caleb

What Is a Quadratic Relation?

A quadratic relation corresponds to a quadratic formula. These formulas are often used to calculate the height of falling rocks, kicked balls or an arched bridge.
A quadratic formula is also sometimes called a second degree formula.

There are THREE quadratic equations:


  1. Vertex Form
  2. Factored Form
  3. Standard Form

How does it apply to everyday life?

Quadratics can be utilized to discover the flight of items. While tossing an object such as a football, the flight of and decent of the ball makes a parabola. Other situations where parabolas can be utilized to figure the path of things are: roller coasters, the path of a basketball being shot or someone kicking a soccer ball. Parabolas can locate:The max height of a ball thrown, how long it takes for the ball to achieve the highest point and what is the initial starting point of the ball before it was thrown.Quadratics are also very important for companies and businesses. Quadratics can show them how to make profit, how much there prices should be and how many customers to expect. Quadratics also helps us build our understanding of measurements of shapes. Once you finish reading this website you will be able to understand quadratics and parabolas very easily.

An Example of a Parabola

This picture labels the vertex and the axis of symmetry on a parabola. The parabola here is opening up because the a is positive. The vertex (also known as the optimal value) in this parabola is the minimum height. If it opened down the vertex would be the maximum height of the parabola. The axis of symmetry is an x coordinate on the graph that splits the parabola in half.
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Unit #1: Vertex Form

Graphing Vertex Forms: Learning Goals

Learning Goal 1: To be able to understand how transformations work

Learning Goal 2: To be able to solve word problems involving quadratic equations

Learning Goal 3: To be able to make a parabola on a graph using vertex form

Summary of Graphing Vertex Form

How to Solve Vertex Form

The equation for the vertex form of a parabola is y=a(x+h)^2+k. If you want to solve with this form you need to substitute y with 0 and solve the equation for x to get the standard form of the equation. If you need to find the y intercept of the equation, substitute the x for 0 and solve to find y.

The Placement of the Parabola

The k and the h in the vertex form determine where the parabola will be on the chart. The k represents the vertical translation of the parabola and the h represents the horizontal translation of the parabola.If the k is positive the vertex of the parabola will be on the upper part of the graph and if it the k is negative, the vertex will appear on the bottom half of the graph. If the h in the formula is positive that means the parabola will be on the left side of the graph, however if it is negative it will be placed on the right side of the graph.

The a determines which way the parabola opens. If the parabola opens downwards, that means that the a is negative. However, if the a is positive, that means the parabola will open upwards

Step Pattern

The step pattern is a good way to find out how to graph a parabola. To do the step pattern you must:

· Start at the vertex and go to the next point on the right.

· Identify the rise and the run from the vertex to that point.

· Write them down.

· Then identify the rise and run to the third point.

· Write them down.

· Keep doing so, until you find a pattern in the run. Then do it on the other side to get the parabola.

Vertex=(h,k)

Axis of Symmetry=(x=h)

Optimal Value (y=k)

Lets Solve a Word Problem

Vertex form word problems

Unit #2: Factored Form: Y= A(x-r)(x-s)

Learning Goals

In this unit you should be able o be able to solve problems using factored form.

You should also try to be able to make a parabola on a graph using factored form.

Summary of Factored Form

The meaning of R, A and S

The R and S of the equation are actually the two X intercepts of the parabola.

The A of the equation determines whether the parabola opens up or down and if it is stretched or compressed. If it opens upwards, the A must be positive. If it opens downwards, the A will have a negative value. If the A has a value that is greater than 1 or less than -1, the parabola will be stretched vertically. However, if the A has a value that is in between 1 and -1, the parabola will be vertically compressed.

Different ways of Factoring

There are 5 different types of factoring, they are:

Greatest Common Factor: Monomial Factoring

When we do monomial factoring we factor out the greatest common factor

5x+10y

=5(x+2y)


Simple Factoring: Simple Trinomial Factoring

To factor a simple trinomial we need 2 numbers that multiply to get the last number, but add up to get the middle number as shown below.

x²+9+20

=(x+5)(x+4)

Complex Factoring: Complex Trinomial Factoring

To solve complex factoring you need to multiply the lead coefficient with the last coefficient. Once you get that number, you must find two numbers that has a product of that number and the sum of the middle number in the trinomial. Once you get those two numbers, replace the middle term with them. After that, just do common factoring.

2x2 – 3x – 35

2x2 -10x +7x -35

2x(x-5) + 7(x-5)

(2x+7)(x-5)


Difference of Squares

x2 -4

(x-2)^2 or (x-2)(x-2)


Perfect Square

x2+12x+36

(x+6)^2 or (x+6)(x+6)

Here is an in depth video about factoring trinomials

factoring trinomials

Axis of Symmetry/Optimal Value

There is an equation to find the axis of symmetry using factored form. To get the axis of symmetry you must add up the R and S values and divide them by 2. The equation looks like this: x=r+s/2. Once you solve this, sub this value into the X to get the optimal value or the vertex.

For example:

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Finding the y intercept

To find the Y intercept, sub x for 0 and solve for Y.

For example:

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Here is a word problem using factoring!!!

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Unit#3: Standard Form: Y=ax² +bx+c

Learning Goals

In this unit you should try to be able to use the quadratic formula and be able to understand the standard form equation. You should also be able to find intercepts using the quadratic formula. Lastly, you should be able to convert standard form into vertex form.

What do the variables mean?

Before you start trying to solve using standard form you must know what the variables in the equation mean. The a in the equation gives you the direction of the opening of the parabola. If it is negative the parabola will open downwards. If the a is positive, however, the parabola will open upwards instead. The value of a will also determine if the parabola will be vertically stretched or compressed. The c in the equation is the y intercept. If the C is negative the y intercept will be on the bottom half of the graph. If the C is positive, it will instead be on the top half of the graph.

The Quadratic Formula

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When do you use the quadratic formula?

You should use the formula when you need to find the x intercepts of a parabola. It is helpful to use this form when you can not use factored form.
Solve Quadratic Equations using Quadratic Formula
The quadratic formula is the formula that use with the standard form, to find the x intercepts of a parabola. Using this formula will easily give you the x intercepts. To use this formula you must sub in the values of the variables.

For Example:

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The two X intercepts are (1.36,0) and (-7.36,0)

Completing the square to get Vertex Form

To convert a standard form equation into a vertex form equation you must follow these rules. Firstly, you need to put the first two terms into a bracket. Secondly, you divide the second term by two then square it. Once you get the value, put the positive version of the value in the brackets after the second term and the negative version out side of the brackets. From here all you have to do is square the binomial.
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Here are some word problems for standard form!!!

Q. A football is tossed from the top of a building and drops to the ground. Its path is represented by the equation: h=t²-4t+7, where h represents the height above the ground in metres and t shows the time in seconds.



a) What is the maximum height of the soccer ball?

b) At what time does the soccer ball reach its maximum height?

c) How tall is the building?

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Reflection

In the end, Quadratics was a truly complex and fun unit to experience. I appreciated taking in the distinctive sorts of strategies to determine word problems. This unit is generally based around diagramming parabola's and it helped me tackle hard word problems proficiently.


When learning vertex form, I learned the characteristics of a parabola like the vertex and y and x intercepts. During the vertex form unit I also learned how create a parabola, name parabola transformations and solve word problems based on quadratics.During the factored form unit, I found out how to expand and simplify quadratic expressions. I also learned different ways of factoring such as complex trinomial factoring, difference of squares, binomial common factoring, perfect square trinomial and factoring by grouping as well as learning to solve factoring word problems. The standard form unit taught me how to find the zeros from the discriminant and how to use the quadratic formula.


This unit has really strengthened my knowledge of graphing, lines and parabolas. Throughout this unit my ability to do word problems has been strengthened and I find graphing much easier. I now also can relate parabolas to real life. I really enjoyed the quadratics unit because of all the new concepts that I have learned.