Quadratics

By: Apinder Sudan

Table Of Contents

Intro

- What is Quadratics

-Linear and non-linear

Factored Form - y = a(x-h) (x-r)

- Zeroes/x intercepts

-Optimal Value

-AOS x=(r+h)÷2

Standard Form - y = ax2 + bx + c

-Axis Of Symmetry

-Zeroes

-Optimal Value

-Completing Square

-Factoring

1.Common

2.Simple trinomial

3.Complex trinomial

4.Perfect Square

5.Difference of squares

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


  • Axis of symmetry (x=h)
  • Optimal Value (y=k)
  • Transformations (translation vertical or horizontal, vertical stretch, reflection
  • Step Pattern
  • X-intercepts or Zeroes (sub y=0 and solve)

What is quadratics?

Quadratic comes from the word "quad" which means square because in the equations, the variable is squared. You can use a quadratic equations to find out and determine many different things such as calculating and graphing the flight path of something like the flight path of a thrown football, or calculating and finding the certain number that will give you maximum profit.

Linear and non-linear

to determine if your equation is linear or non linear, look at the first differences. If the first differences are the same then it is linear, but if the first differences are different then the equation is non linear
What is a Parabola

Factored Form - y = a(x-h) (x-r)

Zeros/X Intercepts

Zeroes are when x is equal to zero.

The X-intercepts are the points or the point at which the parabola intersects the x-axis. A parabola can have either two, one, or zero real x intercepts.

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Axis Of Symmetry (AOS)

Every parabola has an axis of symmetry which is the line that runs down the middle . This line divides the graph into half. The axis of symmetry in factored form uses the formula x=(r+h)÷2. the value of x is the axis of symmetry.

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Optimal Value

The optimal value is the y coordinate of the vertex. Which is the maximum or minimum y point.

To find the optimal value you would need to first find the two x-intercepts and then sub them into the equation (r+h) ÷ 2) and solve.

Graphing a Quadratic Function in Factored Form

Standard Form - y = ax2 + bx + c

Zeros Formula

The quadratic formula can be used to find the roots or zeroes of the equation. It is used for standard form when factoring isn't an option. You will get 2 answers because you have to add and subtract when you're at the plus and minus step.
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Optimal Value

To find the optimal value of any standard form equation first find the axis of symmetry and then substitute it into the equation to find the last value which is the optimal value. The optimal value is the y coordinate of the vertex. The optimal value is the maximum or minimum point on the parabola and is needed to finish the equation.
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Common Factoring

Common Factoring is dividing out unnecessary information and simplifying the equation. After you have multiplied the equation you will end up with the same answer meaning all is left is simplifying. Factoring means dividing and putting it in front of the parentheses. In the equation 9k^3-27k-145 you can see that there is a common multiple of 8 so all you would have to do is put 8 outside the bracket and divide everything else in the equation by 8 and then write the products inside the bracket so

9k^3-27k-145 = 9(k^3 - 3k -18)

Factoring with a simple trinomial


A simple trinomial is an equation that has three different terms, and the value of a is one, in the standard form equation (y=ax^2+bx+c) to factor you need two brackets, and both brackets need to equal to the equation.

Difference Of Squares

Difference of squares is similar to perfect square only instead of the same bracket one of the signs change and the numbers stay the same, because you need one positive and one negative to make the them cancel out

example:

(a+b)(a-b)

a^2-ab+ab-b^2

this equals to a^2-b^2 because a and b get cancelled out each other.

Perfect Squares

Perfect square is when you square root the first and last term, and when you do that you multiply them and then multiply your product by two, which should give you your middle term. IE: 16m^2+24m+9
The first terms square root is 4m and the last terms square root is 3. When you multiply those two you get 12m and multiply that by 2 and you get 24m. So that is a perfect square. Now when you factor it, you should get (4m+3)^2
Quick Way of Graphing a Quadratic Function in Standard Form

Vertex Form

Vertex Form is expressed as the equation:
  • y=a(x-h)^2+k
  • h and k are the points of the vertex so, (h,k)
  • The sign given with the h isn't the right sign, you have to bring "h" to the other side of the equal sign to find out what it is, an example is if it (x-8) it would be x-8=0 next you bring the 8 to the other side changing it sign, so x=8
  • If "a" is positive the parabola would open upwards, a "U" shape, if "a" was negative then you would have a "n" shaped parabola
  • If a<1 the graph widens meaning it is compressed, for example: y=0.5(x-3)^2+7, you would say the parabola is compressed
  • If a>1 the graph stretches meaning it would become narrower, for example: y=2(x-2)^2+2, you would say the parabola would have a vertical stretch of 2
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Quick Way of Graphing a Quadratic Function in Vertex Form

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

-The optimal value is the highest or lowest point of the parabola. The optimal value is represented by the k in the equation.

-If the k value is negative then the vertex will be below the x axis and if the k value is positive then it will be above the x axis

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

-The step pattern is also a very important thing that you need to know.


-The step pattern is the movements that you follow from the vertex to find the other points of the parabola so that you can connect them and create a parabola.


The Pattern:

over 1 up 1

over 2 up 4

over 3 up 9

etc.


-also a key piece of information is that when you are about to use the step pattern you must multiply the up number by your a value so if your a value was 2 then your step pattern would be

over 1 up 1x2

over 2 up 2x2

X intercepts and Zeroes

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

1. 8=2(x+2)^2 Bring 8 over to the other side


2.8/2=(x+2)^2 Divide both sides by 2


3.4=(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

Video

Solving a Equation