Quadratics
By: Apinder Sudan
Table Of Contents
Intro
-Linear and non-linear
Factored Form - y = a(x-h) (x-r)
-Optimal Value
-AOS x=(r+h)÷2
Standard Form - y = ax2 + bx + c
-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?
Linear and non-linear
Factored Form - y = a(x-h) (x-r)
Zeros/X Intercepts
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.
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.
Optimal Value
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.
Standard Form - y = ax2 + bx + c
Zeros Formula
Optimal Value
Common Factoring
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
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
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
Vertex Form
- 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
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.
Optimal Value
-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 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
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