What is a parabola?
- a fixed point
- a fixed straight line
What are second differences?
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
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
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.
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.
This is the effect of multiplying x^2 by a number greater than 1.
This is the effect of multiplying x^2 by a number less than one.
This is the effect of multiplying x^2 by a negative number.
Vertex Form: y = a(x - h)2 + k
The maximum or minimum point on the graph. It’s the point where the graph changes direction.
The highest or lowest value that the parabola takes on.
Axis of Symmetry
The vertical line which cuts the parabola down the middle.
Where the parabola intersects the y-axis.
Where the parabola intersects the x-axis.
X value which makes the equation equal 0.
Graphing with Vertex Form.
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.
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.
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
your axis of symmetry would therefore be 3.5.
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:
In this case the Optimal value would be -2.25.
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!
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!
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.
In the Following equation,
A = 3
B = 4
C = 1
This will help you to understand why the numbers are placed where they are
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.
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.)
Graphing from Standard form
The Different Factoring Methods Are:
-Difference of Squares
You can then use the factored form to easily graph the parabola!
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.
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
Factor the following trinomial.
x2 - 5x + 6
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)
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.
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)
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:
Step 6 (optional): Foil out your answer from Step 5 to check it.
Add the four terms:
2x2 -10x +7x -35 = 2x2 -3x -35
(4 )(4 )
lets try 3 because 3x3 is 9
24 is a negative and 9 is positive so we can use -3 and -3
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
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
Completing the square
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.
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
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
Now collect like terms and you get your vertex form
We have the vertex at (-2,-3)
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.
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
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.
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 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 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
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.