Quadratics 101

By: Jagraj Kang

Table of Contents

Parabolas:

-Vertex Form

-Axis of symmetry

-Optimal Value

-X-intercepts/Zeros/Roots

-Tranformations

-Step Pattern

-First difference

-Second differences


Factored Form:

-Axis of Symmetry

-Optimal Value

-Zeros/ X-intercepts


Standard Form:

-Quadratic Forumla

-Axis of Symmetry

-Optimal Value

-X-intercepts/Zeros


Factoring:

-Common Factoring

-Simple Trinomials

-Complex trinomials

-Completing the squares


Word Problems

What is a Parabola?

A parabola is a curved line that has certain properties such as a axis of symmetry, vertex, y-intercept,zeros(x-intercepts) and etc.
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Vertex Form

y = a(x-h)2+k


  • (h,k) is the vertex
  • if "a" is positive, the parabola will be opening up and if it's negative, then its opening down
  • if "a" is greater than 1 then it will be stretched and if its less than 1, then it will be compressed

Axis of Symmetry

y = a(x-h)2+k


  • "h" is the axis of symmetry. if "h" is positive in the equation, then it is a positive number and vice versa

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Optiaml value

y = a(x-h)2+k


  • "k" is the optimal value
  • if its positive, then it's above the x-axis and if its negative, then it's below the x-axis
  • if its 0, then its on the x-axis

X-intercepts/Zeros/Roots

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

y=4(x+4)^2-16

0=4(x+4)^2-16 Sub Y for 0

16=4(x+4)^2 Bring 8 over to the other side

16/2=(x+4)^2 Divide both sides by 4

4=(x+2)^2 Square both sides by - and +

+-2=x+2 calculate for the X's


-2=x+2

-2-2=x

x=-4


2=x+2

2-2=x

x=0

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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.

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.
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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. But if the second differences are still not constant, it is neither.
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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.

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

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.

Graphing in Factored Form

  1. Find A.O.S
  2. Sub it into the equation to get your y-value
  3. You have the vertex and x-intercepts and just connect the lines
Factored Form - graphs of quadratics

Standard Form y=ax^2+bx+c

Quadratic formula

The quadratic formula finds out the x-intercepts and axis of symmetry which will give you the vertex.
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Solving the Quadratic Formula

Using the Quadratic Formula
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Finding A.O.S

  1. Add both x-intercepts from the Quadratic formula
  2. Then divide it by 2

Optimal Value

  1. Sub A.O.S in the original equation

Discriminant

The discriminant will tell you how many x-intercepts a equation will have without solving the for the entire equation.


  • D>0, 2 x-intercepts
  • D<0, 0 x-intercepts
  • D=0, 1 x-intercept


(discriminant formula is in red below)

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Factoring

Common Factoring

Common Factoring Tutorial

Simple Trinomials (S.F to F.F)

If the trinomial is x^2 + bx + c, where a = 1. This is a simple trinomial.

To solve for a simple trinomial:

  1. See if it is a simple trinomial
  2. If it is, find factors of 6 that can add to 5 (do this for all simple trinomials)
  3. Then put those numbers in (x+_)(x+_)

Example:

0=x^2 + 5x + 6

=2 x 3 = 6

=2 + 3 = 5

y=(x+2)(x+3)


Complex Trinomials

  • 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
  • Add the four terms:
2x2 -10x +7x -35 = 2x2 -3x -35

Factoring Complex Trinomials

Perfect Squares & Difference of Squares

  • Perfect Squares are when you only need one bracket because that bracket squared gives you
  • Difference of squares is similar to perfect square only instead of the same bracket the signs change but the number stays the same.

Perfect Square

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Difference Of Squares

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Linking

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

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Word Problems

A garden measuring 12 meters by 16 meters is to have a pedestrian pathway installed all around it, increasing the total area to 285 square meters. What will be the width of the pathway?

Let x rep. the width of the pathway

x + 12 + x

=2x+12

x + 16 + x

=2x+16


New area

(12 + 2x)(16 + 2x) = 285
192 + 56x + 4x2 = 285
4x2 + 56x – 93 = 0

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Therefore the width of the pathway is 1.5m.
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Work I have done

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Reflection

I think this unit was pretty interesting. There were lots tricks you can do to find answers and multiple methods of finding of answer. These multiple methods got me through most of the unit. If I'm stuck on a question, I'll try to use another method to see if its easier to solve. I found this unit pretty easy as well. I might of not done really good on the tests but that was just due to the factor i might have not studied the day before a test. All in all the unit was pretty fun and good for me and I hope there is more fun on the way