Grade 10 Quadratics 101
By: Himmat Gill
Happy Learning :)
Table of Contents
1. What is a Parabola
2. What are Second Differences
3. How to Graph a Parabola
Part 1# Vertex Form
-Axis of symmetry
Part 2# Factored Form
-Axis of symmetry
How to graph a Parabola
Part 1# Standard Form
-Axis of symmetry
Completing the square to turn to vertex form
Factoring to turn into factored form
- Difference of squares
- perfect squares
REFLECTION ON THE UNIT
1. What is a Parabola?
2. 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)
so the equation is
if everything is zero then the graph's vertex stays at the origin (0,0)
So if you change the K value, it changes the vertex's Y value. If the K is positive the Y will be a VERTICAL TRANSLATION UP. If the K value is negative then Y will be a VERTICAL TRANSLATION DOWN. If you change the H value, it will affect the vertex's X Value. If the H is positive than the X will be a HORIZONTAL TRANSLATION LEFT. (Keep in mind that it is the opposite way so positive H is moving left toward negative and negative H is moving right towards positive.)If the H is negative than the X will be a HORIZONTAL TRANSLATION RIGHT.
Y= a( x-h)^2+k
Axis Of symmetry
To find your Axis of Symmetry from the Vertex Form you would need to look at the H value.
In the equation y= 3(x+2)^2+5 the H value which is 2 turns into a negative and is your axis of symmetry. you would write you axis of symmetry like: x= -2. To find the AOS (axis of symmetry) you would change the sign of what is in the bracket +2 would be -2.
To find the Optimal Value you need to look at the K value because that will be your Optimal Value. In this case y=3(x+2)^2+5 the optimal value will be 5. If the optimal value is a positive number then the vertex will be over the X-Axis but if it is a negative number like -5 then the vertex would be below the X-Axis. If the optimal value is 0 then the vertex will be on the X-Axis
To find the Zeros in the equation then you would need to sub in the Y as 0 then solve for X.
0=2(x+2)^2-8 Sub Y for 0
8=2(x+2)^2 Bring 8 over to the other side
8/2=(x+2)^2 Divide both sides by 2
4=(x+2)^2 Square both sides by - and +
+-2=x+2 calculate for the X's
Vertex Form Graphing
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 weird 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)
5. PART 2 # FACTORED FORM
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!
In the Following equation,
This will help you to understand why the numbers are placed where they are
Axis Of symmetry
6. Part # 3 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
Difference Of Squares
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)
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.
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 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
Thank you for everyone who viewed it!
(Creator of Quadratics 101 site)