Table of Content

What is a parabola

First difference

second differences

Vertex Form

-Axis of symmetry

-Optimal Value

-X-intercepts/ Zeros

-Transformations

-Step Pattern

Factored Form

-Axis of Symmetry

-Optimal Value

-Zeros/ X-intercepts

Standard Form

-Axis of Symmetry

-Optimal Value

-X-intercepts/Zeros

Standard form to factored form.

- Simple Trinomials

-Complex trinomials

-Completing the squares

Parabola

A parabola is a curved line with certain properties. It has an axis of symmetry, a y-intercept and sometimes x-intercepts, and a vertex.
Module D What is a parabola and its parts

First Differences

To find the first difference you must subtract the second Y value from the first. Then you continue doing this for each pair of y-values. If the first differences are constant then it means that the step pattern is linear. Remember to put you x-values in order.

Second differences

For the second difference do the same thing as you did for first difference but this time use the numbers you got from the first difference to find second difference. if it is constant than it is Quadratic. Remember that you only use second differences when the first differences are not constant.

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

There are many things involved with the Vertex Form,

- Axis of symmetry

-Optimal Value

-Transformations

-X-intercepts or zeros

-Step patterns

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.

Find axis of symmetry and vertex of quadratic equation

Optimal Value

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

X-intercepts/zeros

To find the Zeros in the equation then you would need to sub in the Y as 0 then solve for X.

y=2(x+2)^2-8

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

-2=x+2

-2-2=x

-4=x

2=x+2

2-2=x

x=0

Transformations

y=a(x-h)^2+k is the general equation for the quadratic.

where a controls if the parabola opens up or down.

where a controls if the parabola is stretched or compressed.

where h controls the horizontal shift.

where k controls the vertical shift.
All these are used when using transformations.

If A is a positive number then the parabola will be opening up but if it is a negative it will be opening down.

If the A value is greater than 1 then it is being stretched, if it is >1 but <0 then it is being compressed, if it is 0 then nothing changes.

If the H value is increased by a number than the graph will shift up by that many points, but if it is decreased than it will shift down that many points.

If the K value is increased than the graph shifts right by that many points, but if it is decreased than it shifts left by that many points.

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.

How to Graph form Vertex form

To graph from vertex form it is simple all you need to do is first find the vertex. Tom find the vertex all you have to do is look at the equation and see what the k-value is because it is going to be you y on the graph, also you look at your (h-value) which is inside the brackets which is you x value as shown above. Now you use those points to find you vertex.

Next look at the a-value which will tell you how much to move the next point. Now you would just use the step pattern and sub in the a-value. Plot the points on your graph and connect them.

Quick Way of Graphing a Quadratic Function 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.

Zeros/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)

Graphing Factored form

To graph Factored form there are some easy steps you need to remember.

First find the axis of symmetry

Then sub it into the equation to get your y-value

After you have both it will give you the vertex and all that is left to do is connect the points on your graph

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

Standard Form y=ax^2+bx+c

You can use the quadratic formula to find out the x-intercepts and axis of symmetry which will give you the vertex.

X-intercepts

To give you an idea of how to use the quadratic formula to find the x-intercepts look below to find how to solve for the equation x^2+2x-3

Axis of Symmety

In order to find the Axis of Symmetry you have to use the Quadratic formula or using the x=-b/2a

In the following you use the formula -b/2a when finding the axis of symmetry.

First -b= -4 and 2a= 2*2 which is 4

-4/4 is your axis of symmetry

-1 is your axis of symmetry

Finding the Vertex and Axis of Symmetry.wmv

Optimal Value

To find the optimal value you would plug in the axis of symmetry into the original equation and solve for y also known as your Optimal Value.

Example: y=2x^2-4x-3

y=2(-1)^2-4(-1)-3

y=2+4-3

y=3

Therefore your optimal value is 3.

Discriminant

The discriminant gives you some information you may not have know before like how many x intercepts the equation will have.

The discriminant is a part of the quadratic formula as seen below.

If the discriminant is greater than 0 than there will be 2 x-intercepts

If the discriminant is equal to 0 than there will be 1 x-intercept

If the discriminant is less than 0 than there will be no x-intercept

Using the Discriminant for Quadratic Equations

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

Simple Factoring

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

x^2-7x+10

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

-7x is B

+10 is C

two factors that add to -7 but also multiply to +10.

The factors could be 2 and 5.

-2 and -5 because C is a negative.

so the answer will be the following

(x-2) (x-5).

Standard form to Factored form- Simple Trinomials

To turn an equation from standard form to factored form you must first see if it is a complex trinomial or a simple trinomial. To determine whether it is complex or not you must look to see if the x^2 has a coefficient in front of it. If the x^2 does not have a coefficient in front then it is a simple trinomial. To solve a simple trinomial you would simply look at the multiples of the C value which when added together equal to the B value and multiplies to the C value:

y=x^2+4x+3- so the multiples of 3 are 1, and 3

(x+1) (x+3) -simply put the multiples into the brackets along with the x's

Complex trinomials using 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

Complex trinomials using 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)

Complex

Perfect squares

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

25x^2-30x+9

First we can see that 25 and 9 are both perfect squares

Since we see this the possibility of it being a perfect square trinomial is likely

One thing to remember is that for perfect squares there are 2 rules you must follow

1) The first term and the last tem have to be perfect squares

2) The two squares multiplied by each other than by 2 has to equal to the b-value.

Lets see if this works:

The square of 25 is 5

The square of 9 is 3

Now if we multiply those we get 15 and 15*2=30 so this works.

Lets solve this equation:

(5x-3) (5x-3)

As you can see it does indeed work.

Lets check our work now:

5x*5x= 25x^2

5x*-3=-15x

-3*5x= -15

-3* -3= 9

Therefore its works and is 25x^2-30x+9

You could also solve this as (-5x+3) (-5x+3)

Either way works

Factoring Perfect Square Trinomials - Ex 2

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

All three forms: Vertex, factoring and standard can be changed to one another as shown below.

Also:

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 unit has been an interesting experience for me. Every day we learned something new and challenging. During the beginning of every new part of quadratics has been difficult to me and hard to understand but as we progressed into the unit Ms. Johal has helped me to understand all of the parts and units of quadratics in a easy and fun way. Whenever I got bad on a quiz in quadratics I would always get discouraged but Ms. Johal and my peers helped me to overcome it. Ms. Johal always was more than willing to assist me in my time of need and always explained everything clearly to make me understand which reflected on my test that followed. All in All it was a great experience learning the unit of Quadratics.