Quadratics 101

Learn about quadratics! By: Farhaan G.

Table of Contents

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

What is a Parabola?

A parabola is a curved line that has specific properties; they have an axis of symmetry, a y-intercept, a vertex, an optimal value, a maximum or minimum, and sometimes it has x-intercepts.

The axis of symmetry is a the vertical line that passes through the vertex - it is written as (x value of vertex, 0).

The y-intercept is the point where the parabola passes through the y-axis.

The zeroes, also known as the x-intercepts, are the points where the parabola passes through the x-axis; however, not all parabolas have zeroes.

The optimal value is the horizontal line that passes through the vertex - it is written as (0, y value of vertex).

The maximum or minimum is the y value of the vertex; it is a maximum if the parabola is negative, and a minimum if the parabola is positive.

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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 in the table of values; continue doing this for each pair of y-values.

If the first differences are constant, then the step pattern is linear. Remember to put your x-values in order so it makes sense.

Second Differences

For finding second differences do the same thing as you did for finding first differences; but this time, use the numbers you got from the first differences to find second differences. if the second differences are constant, then it is Quadratic.

Remember: Only use second differences when the first differences are not constant.

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Vertex Form

The vertex form is the easiest to use when you are looking for the vertex of a parabola.
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Axis of Symmetry

To find your Axis of Symmetry from the vertex form, you need to look at the H value.

In the equation y= 4(x+8)^2+7, the H-value, which is 8, is multiplied by (-1). This is your axis of symmetry (x-value of vertex). For this example, you would write your axis of symmetry as: x= -8.

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 is your optimal value. In this case y=4(x+8)^2+7, so the optimal value is 7. If the optimal value is a positive number, that means the vertex will be over the x-axis, but if it is a negative number, then the vertex will be below the x-axis. If the optimal value is 0, then the vertex will be on the x-axis.


Zeroes are also known as x-intercepts.

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








y=a(x-h)^2+k is the vertex form equation.

Whether a is positive or negative controls if the parabola opens up or down.

The value of a controls if the parabola is stretched or compressed.

The value of h controls the horizontal shift from the origin.

The value of k controls the vertical shift from the origin.
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 less than 1 but greater than 0 then it is being compressed.

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.

4.1 Quadratic Functions - Transformations
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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.
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How to Graph from 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.

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Quick Way of Graphing a Quadratic Function in Vertex Form

Factored Form

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

Factored form is good for finding the zeroes quickly.

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.

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:





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)

How to Graph 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

Quadratic Formula

You can use the quadratic formula to find out the x-intercepts and axis of symmetry which will give you the vertex.
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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.
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How to Graph From Standard Form (My Video)


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.


Simple Factoring

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


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

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

If you want to check your answer simply expand both brackets.

Complex Trinomials: 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:


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

Add the four terms:

2x2 -10x +7x -35 = 2x2 -3x -35

Complex Trinomials: Trial & 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)


Perfect Squares

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


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


-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.




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


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.

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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.

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Using the Discriminant for Quadratic Equations

Summary of Linking Forms and Using Them Effectively

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


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.

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Throughout Quadratics, I feel like I have actually learned a lot, and I am able to say that this unit increased my overall understanding of using equations to solve problems, because there was no other way to do so. I did all my homework, and completed classwork on time, so I had the practice needed to do well. However, I still was not good enough for my own satisfaction, as I was making unnecessary mistakes that could easily be fixed through a quick scan over my work. So, in this unit, a main learning skill I have developed is looking over my work after I have finished it, because, in doing so, I fix my own mistakes.