Quadratics

101

Table of Contents

Table of Contents

1. First and Second Differences

2. Parabolas

3. Vertex form

4. Graphing from factored form

5. Expanding

6. Multiplying Binomials

7. Common Factors

First & Second Differences

First & Second Differences

To tell if a relation is linear or not, you must look at the first differences. If the first differences are the same throughout the relation then that means the the relation is linear.


Second differences are used to tell if a relation is quadratic or not. In a quadratic relation the first differences don't have to be the same, but the second differences must.




Parabolas

A very basic quadratic relation is y=x^2

A parabola is a symmetrical curve (both sides of the parabola are the same)


Properties of Parabolas:

Vertex- The vertex is the lowest point of the parabola and is located in the center parabola

Axis of Symmetry- A vertical line that divides the parabola in half perfectly

Y intercept- The point where the parabola crosses through the Y axis

X Intercept- The point where the parabola crosses through the X axis

Min/Max Value- The Y value of the vertex


A quadratic relationship can be graphed by using a chart to organize the x and y as shown above



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

There are many things that you can find with vertex form such as


- Axis of symmetry

-Optimal Value

-X-intercepts or zeros

-Transformations

-Step pattern



Axis of Symmetry

To find the axis of symmetry from vertex form you should look at the H value. In the equation y=4(x+3)^2+9 the H value (3) will turn negative and is your axis of symmetry. You would write the axis of symmetry as x=-3.


Optimal Value

The K value represents the optimal value. In the equation y=4(x+3)^2+9 the optimal value would be 9. The optimal value is the Y coordinate of the vertex.


X Intercepts/Zeroes

To find the X intercepts of a parabola using vertex form you need to sub in the Y value as 0 and solve for X. Here is an example.

y=3(x+3)^2-12

0=3(x+3)^2-12 Sub Y in as 0


12=3(x+3)^2 Take 12 to the other side of the equation

12/3=(x+3)^2 Divide both sides by 3

4=(x+6)^2 Square booth sides by - & +

-+2=x+6 Now calculate for the zeroes


-2=x+6

-2-6=x

-8=x


2=x+6

2-6=x

-4=x


Transformations

All variables in the vertex form equation y=a(x-h)^2+k are used in transformations


  • a controls if the parabola opens up or down and how stretched/ compressed the parabola is
  • h controls the horizontal shift
  • k controls the vertical shift



If the A value is positive then the parabola will open upwards, if it is negative then it will open downwards.

If the A value is greater than one then the parabola will be stretched. If it is smaller than one but greater than zero it means the parabola is compressed.

The H value represents the horizontal position of the vertex in relation to the origin. If the h value is negative then the vertex shifts to the right, if it is positive the vertex shifts to the left.

The K value represents the vertex of the parabola on the Y axis.


Step Patterns

Step patterns are very important for graphing. The step pattern is a simple rule that will help you create a parabola. The most basic step pattern is 1:1, 2:4, 3:9 and so on. So for a basic parabola with an A value of one you would move right to the vertex once and then up/down once, then right 2 and up/down 4. You would also mirror this on the other side of the parabola.


Graphing From Vertex Form

Graphing from vertex form is simple. First you need to find the vertex. To find the location of the vertex you must look at your h value (which is the x coordinate of the vertex) and your k value (which is the y coordinate of the vertex). Using that information you plot the vertex. Now you would look at your a value to tell if the parabola opens up or down. Next you use the step pattern. Plot the points and connect them.


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



Axis of Symmetry

To find the axis of symmetry using factored form you must find the middle point between both of the zeroes. To do this you must take the numbers outside of the brackets and change their sign. For example in y=(x-4)(x-8) the -4 and -8 will become 4 and 8. Then, to find the axis of symmetry you add both numbers and divide them by two.

4+8=12

12/2=6

The axis of symmetry for this equation would be 6.


Optimal Value

To find the optimal value in factored form you would have to sub in the axis of symmetry as x and solve for y.

y=(x-4)(x-8)

y=(6-4)(6-8)

y=(2)(-2)

y=-4

The optimal value would therefore equal -4 in this example.


Finding X intercepts/zeroes

In factored form all you have to do to find the zeroes is take the numbers outside of the bracket and change the signs in front of them. In y=(x-4)(x-8) the zeroes/x intercepts would be 4 and 8.


Graphing from Factored Form

Graphing in factored form is simple.


  • Find the axis of symmetry
  • Sub it into the equation to find the y value of the vertex
  • Plot the vertex & zeroes then connect them


Standard Form y=ax^2+bx+c



Common Factoring

Common factoring can be used to factor out any number or variable that all of the terms can be divided by. Doing this simplifies your equation. For example in 10xy+40x+30x^3y all of the terms are divisible by five and x. That equation would become 10x(y+4+3x^2y).


Simple Factoring

To simple factor you need to take the equation and split it up into two brackets that equal the original equation when multiplied together. For this example we will use x^2+6x+8. To properly factor you must find two numbers that add up to the B but when multiplied equal the C. x^2+6x+8 when factored is (x+4)(x+2)


Factoring Complex Trinomials

Example:

Factor the following trinomial.

2x^2 +10x+12

Solution:

Step 1:The first term is 2x^2 , which is the product of 2x and x. Therefore, the first term in each bracket must be x.

2x^2+10x+12


Step 2: First you want to simplify the equation by common factoring

2x+10x+12

=2(5x+6)


Step 3: Next you should look for factors of the two numbers in the brackets in this case those numbers are two and three.

2(5x+6)

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

Therefore the factored form of 2x^2+10x+12 is 2(x+2)(x+3)


Perfect Squares



Differences of Squaresdds