What is a parabola?

A parabola is a curve where there is any point that is at an equal distance from:
• a fixed point
• a fixed straight line
Module D What is a parabola and its parts

What are second differences?

If you look at the graph, the differences between the table are called first differences, if they are all equal, you can figure out that the table is a linear graph, the differences from the first differences are the tricky part. As you see in the graph, the second differences are the same, therefore this means that if graphed, it would be a quadratic relation. This means that it would be a parabola.

Easy rule to remember:

1. Same first differences = linear graph ( Straight Line)

2. Same second differences = quadratic graph ( Parabola/ Curved Line)

Closer look at the equation

Y=-2(X+2)2-3

The negative in front of the 2 means that the parabola will open downward.
The 2 in front of the bracket means the step pattern will be multiplied by that number.
The +2 means the parabola will go 2 spaces to the left.
The -3 means that the parabola will go down 3 spaces.

Whats the step pattern?

Regular Step Pattern

Over One, up Over Two, up

1 1

2 4

3 9

How do we know this? - Well we just sqaure the amount that is going over, for example over 5, will result in up 25.

What does multiplying x^2 by a number mean? (The number before the bracket)

The number before the bracket is what the step pattern will be multiplied by, for example the equation ontop where the step pattern is multiplied by 2. This will result in over one, up 2, over two, up 8, over three, up 18 and so on. Another example can be 0.5, which will result in over one, up 1/2, over two, up 2.

Transformations

Vertical Translation (upward)

This is the effect of adding a number to x^2 after the square.

Vertical Translation (downward)

This is the effect of subtracting a number from x^2 after the square.

Horizontal Translation (Left)

This is the effect of adding a number to x inside the bracket.

Horizontal Translation (Right)

This is the effect of subtracting a number from x inside the bracket.

Vertical Stretch

This is the effect of multiplying x^2 by a number greater than 1.

Vertical Compression

This is the effect of multiplying x^2 by a number less than one.

Vertical Reflection

This is the effect of multiplying x^2 by a negative number.

Quadratic Functions - Transformations in Vertex Form

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

The vertex of a parobala is the max or min point on the graph. The point where the graph changes direction. So, on y= a( x- h)2 + k, ( h,k ) is the vertex on the parobala.

Definitions

Vertex

The maximum or minimum point on the graph. It’s the point where the graph changes direction.

(X,Y)

Optimal Value

The highest or lowest value that the parabola takes on.

Y=____

Axis of Symmetry

The vertical line which cuts the parabola down the middle.

X=____

Y intercept

Where the parabola intersects the y-axis.

(0,__)

X intercept

Where the parabola intersects the x-axis.

(__,0)

Zeros

X value which makes the equation equal 0.

X=____

Graphing with Vertex Form.

Vertex:

A key thing to know is that the reason Y= a( x-h)^2+k is called vertex form because you already have the vertex/optimal value given to you. the H value is the (x) and the K value is the (y). One thing to remember is that for some reason, the H value is always opposite, for example: if Y= a(x-6)^2+ 5 is the equation, the vertex would be (6,5) not (-6,5) because the -6 has to be opposite therefore the sign changes to positive.

Step Pattern:

Another key thing to know is the step pattern. The step pattern is a basic rule to help you create your parabola. The step pattern is 1:1, 2:4. This means that from the vertex to create the parabola you have to use the step pattern. You have to move left or right 1 and then up or down 1, followed by left or right 2 and then up or down 4. The relation is to square the left right number. Thus if you move left or right 3 of the vertex you know that you have to go up or down 9. This pattern continues on and on.

Graphing a parabola in vertex form

Axis of symmetry

To find the axis of symmetry (AOS) you must first take the numbers inside the brackets out. y=(x-2)(x-5) to take out the coefficients you must first change the signs. -2 will become 2 and -5 will become 5. Next we must add the two number and divide it by two 2+5

=7/2

x=3.5

your axis of symmetry would therefore be 3.5.

Optimal Value

To find the optimal value from factored form you would sub in the axis of symmetry into the x's that are in the brackets and solve for y:

y=(x-2)(x-5)

y=(3.5-2)(3.5-5)

y=(1.5)(-1.5)

y=-2.25

In this case the Optimal value would be -2.25.

X-intercepts

In factored form it is very simple to find the zeros all you need to do is take the coefficients from inside the bracket and change the sign and you would have your zeros.

y=(x-2)(x-5) in this equation you would take the -2 and -5 out of the bracket and change the sign which would make it 2 and 5 which are you two x-intercepts(Zeros)

Another key thing to know is that the reason that even if factored form doesn't give you the vertex of the parabola, if gives you another key piece of information known as the X-intercepts. The two X's in the brackets are the X intercepts, again making sure that they are the opposite signs! This is helpful because is in factored form, you don't need to use the Step Pattern. All you do is take the given X intercepts to find the Vertex as described in the video below, and then draw the parabola!

How to find Vertex?: Ex. Y= 2(x-3)(x+5)

To Find the Vertex there are 5 simple easy steps. First, you take the X intercepts. So for the Example Equation the X Intercepts would be +3 and -5 because you have to switch the signs! Next you have to add the X-Intercepts so +3+(-5)= -2 So that gives you the X value for the Vertex. It is also the middle of the Parabola. Now you sub in X into the equation and solve for Y!

Y= 2(-2-3)(-2+5)

Y= 2(-5)(3)

Y= -10(3)

Y= -30

And now you have the Y as well. (-2,-30) is the Vertex of this equation: Y= 2(x-3)(x+5)

Now that we have the Vertex and the two X intercepts, all you have to do is connect the dots and make a parabola from the X intercept and Vertex!

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

Standard Form

Quadratic formula is used to figure out the x-intercepts and axis of symmetry and also the vertex when you have a parabola in standard form.

X-intercepts

To find The X intercepts

In the Following equation,

A = 3

B = 4

C = 1

This will help you to understand why the numbers are placed where they are

Finding the x-intercepts using the Quadratic Formula

Axis of symmetry

To find the axis of symmetry you have to use a part of the Quadratic formula which is the x=-b/2a part.

Optimal Value

Use the x=-b/2a part of the quadratic formula to find the Axis of Symmetry then sub in X to find the Y(Optimal Value.)

Find axis of symmetry and vertex of quadratic equation

Graphing from Standard form

Graphing straight from Standard Form is a bit hard so we try to convert Standard form into Factored Form to Graph. There are 5 different factoring methods for Standard form depending on how the equation is written down. We will discuss all the different methods, and explain as well.

The Different Factoring Methods Are:

-Common Factoring

-Simple Trinomials

-Complex Trinomials

-Perfect Squares

-Difference of Squares

You can then use the factored form to easily graph the parabola!

Common Factoring

Common factoring can be used to factor out any number or letter that all of the terms are divisible by. By doing this you can simplify your equation. This can be used in standard form as well.

5xy+10y+45x^2y^2- as you can see all of the terms are divisible by 5 and y

5y(x+2+9x^2y)- this is what you would get by factoring the equation by common factoring

For further detail look at the video below.

Common Factoring

Simple Factoring

To Simple Factor, it's not hard. All you have to do is understand the equation.

x^2-2x-35

to factor you need two brackets, and both brackets need to equal to the equation.

Easy rule to follow is multiply to C and add to B.

x^2 is A

-2x is B

-35 is C

two factors that add to -35 but also multiply to -2.

The factors could be 7 and 5.

-7 and +5 because C is a negative.

so the answer will be the following...

Trial and error

Example:
Factor the following trinomial.
x2 - 5x + 6
Solution:
Step 1:The first term is x2, which is the product of x and x. Therefore, the first term in each bracket must be x, i.e.
x2 - 5x + 6 = (x ... )(x ... )

Step 2: The last term is 6. The possible factors are ±1 and ±6 or ±2 and ±3. So, we have the following choices.
(x + 1)(x + 6)
(x - 1)(x - 6)
(x + 3)(x + 2)
(x - 3 )(x - 2)
The only pair of factors which gives -5x as the middle term is (x - 3)(x - 2)
Step 3: The answer is then
x2 - 5x + 6 = (x - 3 )(x - 2)

Decomposition

Step 1: Multiply the lead coefficient (2 in this case) by the constant term (-35 in this case) to get -70.
Step 2: Find two numbers that multiply to make the product from step 1, but add to make the middle term coefficient (-3, in this case). Therefore, for our example, we need find the two numbers that multiply to make -70 but add to make -3. Of course, the numbers are -10 and 7.
Step 3: Rewrite the original trinomial, replacing the middle term with two terms whose coefficients are the numbers from step 2.
In other words,
2x2 -3x – 35 becomes
2x2 -10x +7x -35.
Step 4
Common factor the first two terms from step 3. Then, common factor the last two. Do the pairs separately; it won’t be the same common factor for the first two as for the last two.
2x(x-5) + 7(x-5)
Step 5
Notice from Step 4 that, although the common factors you took out front don’t match, the brackets do match. Put the common factors in their own bracket, then rewrite:
(2x+7)(x-5)
Step 6 (optional): Foil out your answer from Step 5 to check it.
First: 2x(x)=2x2
Outer: 2x(-5)=-10x
Inner: 7(x)=7x
Last: 7(-5)=-35
2x2 -10x +7x -35 = 2x2 -3x -35
Factoring Trinomials (A quadratic Trinomial) by Trial and Error

Perfect Squares

Perfect Squares are when you only need one bracket because that bracket squared gives you the equation. For Example:

16x^2-24x+9

(4 )(4 )

lets try 3 because 3x3 is 9

24 is a negative and 9 is positive so we can use -3 and -3

(4x-3)(4x-3)

4x.4x= 16x^2

4x.-3= -12x

-3.4x= -12x

-3.-3= 9

16x^2-24x+9

so (4x-3)(4x-3) works but because both the brackets are exactly the same, we can square the brackets thus the factored form for 16x^2-24x+9 would be (4x-3)^2

Factoring perfect square trinomials

Difference of Squares

Difference of squares is similar to perfect square only instead of the same bracket the signs change but the number stays the same.

example:

(a+b)(a-b)

a^2-ab+ab-b^2

this equals to a^2-b^2 because ab get cancelled out with each other. This tells us that for difference of squares we need one positive and one negative. This concept is further explained in the video shown below

Factoring difference of squares

Completing the square

Completing the Square is a method to help you turn standard form into vertex form.

As you may know, Y=a(x-h)^2+k is a perfect square and Standard form Y=ax^2+x+b doesn't. So all completing the square does is make standard form look into a perfect square.

Example:

lets Take Y=2x^2+8x+5

Start of by blocking the 2x^2+8x

so it looks like (2x^2+8x) +5

Factor out 2x^2 because in vertex form, X doesn't have a coefficient

so Y=2(x^2+4x)+5

Now we want to make (x^2+4x) a perfect square we need to add another term there.

so it will be Y=2(x^2+4x+_____)+5 and so to get that term we have a formula

x/2^2 so basically 4/2=2^2=4 so it would look like Y=2(x^2+4x+4)+5 but just adding a number into an equation isn't allowed so you need to balance the equation by adding a -4. So now the equation looks like Y=2(x^2+4x+4-4)+5

so Y=2(x^2+4x+4-4)+5 is a perfect square of (x+2)^2 therefore Y=2((x+2)^2-4)+5

so now you can expand Y=2(x^2+4x+4-4)+5 thus you get

Y=2(x+2)^2-8+5

Now collect like terms and you get your vertex form

Y=2(x+2)^2-3

We have the vertex at (-2,-3)

Completing the square

Discriminants

So the Quadratic Formula as we remember is used to find out information from the Standard Form, but quadratic formula also gives us more information which we may not have known.

So as you see on the top half, there is a square root, what whatever is inside a square root is called the discriminant.So the Discriminant is the b^2 -4ac. The Discriminant tells us how many quadratic solutions there are for the requested standard form equation.

1. If the Discriminant is a negative, then there are no solutions

2. If the Discriminant is 0, then there is only one solution

3, If the Discriminant is above 1, then there are two solutions

Connections

Vertex Form:

Vertex form can be converted to factored form by subbing Y=0 and then find find the factored form.Vertex Form can also be converted into standard form by expanding and simplifying the equation.

Graphing Vertex Form:

Vertex form us the vertex which we is helpful to graph. This is already given to us. Next we use the step pattern to graph the parabola.

Factored Form:

Factored form can be turned into vertex form by finding the vertex using the axis of symmetry and optimal value and then subbing those into the vertex form equation. It can also be turned into standard form by expanding and simplifying the equation.

Graphing Factored Form:

Factored form is the easiest to graph because the x intercepts are given. all you do is find the AOS and then the optimal value to graph the parabola by connecting the points.

Standard Form:

Standard form can be converted into vertex form by using the completing the square method. It can be converted into factored form by using the factoring methods seen above. Common, Simple, Complex, Different of squares, Perfect squares.

Graphing Standard Form:

Graphing Standard Form is hard so there are two ways to graph. Use the quadratic formula to find X intercepts or Complete the Square to get vertex form to graph or factor to turn into factored form then graph

Reflection

This Quadratics unit has been an exciting experience for me in the grade 10 school year. Math has always been one of my favourite subjects and i grow to like it even more. I feel like this is where you can actually find math helpful of life. I have been doing fairly well in this unit and I understand it, partially because Ms. Molyneaux is a wonderful teacher. She understands all the possible questions we as learners have and she is always there to help. Most my tests were done well but I tend to make careless mistakes and often forget things on the spot. Other than that i feel like I am doing good in math. But like always, there is always more room for improvement and i will always try to work harder, one thing I know I have to work on is completing homework since homework helps you to better understand the topics. I feel like other than that, I am doing well in math. I will continue to work hard and try my best to maintain the highest mark possible from my fullest potential.