Quadratic

Relation, Expression and Equations

About Quadratics

Quadratic is the largest unit in MPM2Do and the hardest if you don't keep up with your work. This website will help you understand different topics in this unit thoroughly.

So get ready with your pens and notebooks :D

Table of Context

  1. Introducing the Parabola.
  2. Analyzing quadratics
  3. Graphing from vertex form
  4. Transformation.
  5. Finding the equation given vertex
  6. Graphing from factored forming
  7. Expanding
  8. Common factoring
  9. Factoring by grouping
  10. Factoring simple trinomials
  11. Factoring complex trinomials
  12. Factoring special trinomials
  13. Video on Factoring
  14. Completing the Square
  15. Solving by isolating x
  16. Discriminant
  17. Quadratic Formula
  18. Shape problems
  19. Revenue problems
  20. Motion problems
  21. Consecutive intergers
  22. Reflection
  23. Links between topics

Introducing....The Parabola

The graph of a quadratic relation is called a parabola. The parabola has some important features such as:
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Main ideas:


  • Parabolas can open up or down
  • The zero of a parabola is where he graph crosses the x-axis
  • Zeroes can also be called x intercepts or roots
  • The axis of symmetry divides the parabola into equal halves
  • The vertex of a parabola is the point where the axis of symmetry and the parabola meet.
  • The optimal value is the value of the y co- ordinate of the vertex
  • The y intercept of a parabola is where the graph crosses the y axis.

Analyzing Quadratics.

Recall that linear relationships have equal first differences. Now: Quadratic relationships have equal second differences.

For example:

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The second differences are the same therefore it is a quadratic equation.

Graphing from vertex form

First of all, you must memorize the vertex equation which gives you a lot of information on how to graph which is:

y = a(x - h )2+ k

In order to graph from vertex form, you need to know what each variable represent:

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To graph a parabola, you also need to know how to use a step pattern.

For example:


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As you can see on the above image, the parabola opens upwards. There are cases were the a value is greater or less than 1 which affect the step pattern.

For example:

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Conclusion

  • The original step pattern of a parabola is over one up one, over two up four.
  • The vertex decides where we start the step pattern.
  • The 'a' value decides if the step pattern changes (it multiples the vertex part of the step pattern

Graphing transformations of Parabolas

The below image summarizes the transformations of graphing in vertex form as well as the step pattern which are extremely important to know in this unit.
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Finding the equation given vertex

This is important since it helps you find the value of a.
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Graphing from factored form

To begin with, a new equation is introduced in this part of the unit:

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

You need to know how each variable affects the graph before we start learning how to graph

1. a - causes the vertical stretch or compression. If a>1 it is a vertical stretch. If -1<a>1 its is a vertical compression.

- if the parabola face up or down. If a is negative, the parabola face down. If a is positive, the parabola faces up.

2. r - first x-intercept

3. s - second x-intercept


Below is a YouTube video to help you understand how to graph a parabola from factored form:

Graphing Factored Form of Quadratic Functions

Expanding

Expanding the brackets is really important in this unit.

The example below explains how expand an equation

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Common Factoring

"Factors are the numbers you multiply together to get another number

When you find the factors of two or more numbers, and then find some factors are the same ("common"), then they are the "common factors".'

(https://www.mathsisfun.com/definitions/common-factor.html)

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Remember to always check your equations if you can common factor them before solving the equation

Factoring by grouping

Factoring by grouping means that you will group terms with common factors before factoring.
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Factoring Simple Trinomials

In simple trinomials, the a value = (+/-) 1 always.

Equation- x2 + bx + c

Step 1: Find multiples of x2 and c


Step 2: Form two brackets

Step 3: Insert the multiples of x2 and c in both of the brackets

Step 4: Expand the brackets to check if you got the right answer


Below is an example of simple trinomials question.

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Factoring Complex Trinomials

In complex trinomials, the a value unlike simple trinomials is greater than 1.

Equation - ax2 + bx + c

Step 1: Find multiples of ax2 and c


Step 2: Form two brackets

Step 3: Insert the multiples of ax2 and c in both of the brackets

Step 4: Expand the brackets to check if you got the right answer


Below is an example of complex trinomials

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Factoring Special Trinomials

Perfect squares - a value and z are squares (x+z)2

Step 1: Find the multiples of a and c

Step 2: Multiply the numbers by 2 to get the middle number

Step 3: Insert the multiples of the squares in the equation above

Different of squares - ( x + z ) ( x - z )

Step 1: Find the multiples of x and c

Step 2: Insert both the numbers you got in the brackets one + and the other -


It is very important to know the difference between them. Below is an example of this equations

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Video on Factoring

Factoring

Completing the square

Completing the square is where we change standard form (y= ax2+bx+c) to vertex form (y= a(x-h)2+k).


Example:

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Steps

  1. Remove the common factor form x2 and x term -coefficient
  2. Find the constant that must be added and subtracted to create a perfect square
  3. Group the 3 terms that form the perfect square. (move the subtracted value outside the bracket b multiplying it by the common factor first)
  4. Factor the perfect square and collect like terms



Solving by isolating for x

This is a very important part of the unit because it helps you find the x-intercepts.

Below is a step by step example that shows you how to answer questions like this.

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Discriminant

The discriminant is the name given to the expression that appears under the square root (radical) sign in the quadratic formula.


Discriminant - b2 - 4ac


It quickly tells you the number of real roots, or in other words, the number of x-intercepts, associated with a quadratic equation.


D>0 - 2

D<0- 0

D=0- 1


Below is an example of how to find a discriminant

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Quadratic formula

The Quadratic Formula is a very important part the quadratic units.

A new formula is introduced which is:

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This formula is important because it helps you find the x-intercepts.

Below is an example of a quadratic formula question

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Shape problems

Shape problems mostly consist of Triangles, rectangles and squares.

The questions mostly asked is to find the area, perimeter and missing sides.


Below is an example about a right triangle

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This problem goes back to standard form equation and also to expanding.

Revenue Problems

Revenue problems are mostly about money and increase. In this word problem, you must provide let statements in order to get full marks.


Below is an example of a revenue problem

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In this equation, there is more than 1 question that is asked in the end of the word problem. Just as I did above, make sure to answer both of them

Motion Problems

Motion problems are about distance, time and speed.

Example:

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Consecutive Integers

Consecutive integers are problems that are about numbers that are next to each other for example- 1 and 2.


Below is an example of a consecutive integers

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REFLECTION

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This is one of the questions i did in a quiz. I didn't know everything about transformation which got me half the mark in the question. I could have done been more specific like how it goes up 5 units.
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In this question i forgot to common factor the equation. What I could have done is check each equation before I do it to see if you could factor it or not. Just by forgetting to do this, I got zero on this question.

Links between topics in Quadratics

Factoring connects to Graphing

  • Factoring helps you understand equations such as complex trinomials and helps you know what each variables does while graphing
  • It also helps you know what the x-intercepts are and where the vertex is by calculating

Standard form connects to Vertex form

  • Both topics are linked back together through the topic of Completing the square
  • You convert standard form to vertex form in order to get the vetex

Discriminant connects to Quadratic Formula

  • The discriminant equation goes back to half of the quadratic equation

Simple trinomials connects to Complex trinomials

  • Simple and Complex have the same equation except for how the a-value is equal to 1 in simple but in complex, the a-value equals to more than 1