Quadratic Relations

Gurjot Parmar

Contents

Vertex Form:

- Axis of symmetry

- Optimal Value

- Transformations

- X intercepts

- Step pattern


Factored Form:

- X intercepts

- Axis of symmetry

- Optimal value


Standard Form:

- Zeroes

- Axis of symmetry

- Optimal value

- Completing the square

- Factoring to turn to factored form

  • Common
  • Simple trinomial
  • Complex trinomial
  • Perfect squares
  • Difference of squares

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

Transformations

In Vertex form the general equation is y = a(x - h)^2 + k. In which each variable is responsible for a transformation.


'a' controls whether the parabola opens up or down (if the 'a' value is negative it opens down, if the 'a' value is positive it will open up)


'a' also controls whether the parabola is stretched or compressed


'-h' controls the horizontal shift (left or right)


'k' controls the vertical shift (up or down)

Step Pattern

The step pattern for y = x^2 :


y = x^2

y = 1^2

y = 1 x 1

y = 1


This means that we move one unit over and one unit up (1,1).


y = x^2

y = 2^2

y = 2 x 2

y = 4


This means that we move 2 units over and 4 units up (2,4)

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Axis of Symmetry

The axis of symmetry is the line of symmetry for a graph. Both sides of the parabola are equal on ether side of the line. In vertex form the 'h' variable is the axis of symmetry.


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

Optimal Value

In the vertex the optimal value is "y". The optimal value is the maximum or minimum value depending on whether your parabola is facing up or down.

How to find x-intercepts for vertex form

First sub y = 0 into the equation y = (x-3)^2 -16.


0 = (x-3)^2 -16


Then bring the -16 to the other side and square root both sides to simplify.


√-16 = √(x-3)^2

4 = x-3


Now add and subtract the (-3) from 4 to find the x intercepts.


4+3= 7

-4+3= -1


This tells us that the x intercepts are 7,0 and -1,0.

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

How to graph factored form

To graph factored form you have to find the x intercepts with the x intercepts you can than use the formula r+S/2 to find the AOS. Then you plug in the AOS into the equation to find the optimal value.


y = -3(x-4)(x+2)


First you want to find the x intercepts by making the brackets equal to zero.


x-4 = 0 x+2 = 0

x = 4 x = -2

(4,0) (-2,0)


After finding the x intercepts you want to plug use the equation r+s/2 to find the Axis of Symmetry.


= -4+2/2

= -2/2

Axis of symmetry= 1


Now you want to plug in the AOS (1) into the equation to find the optimal value.


y = -3(x-4)(x+2)

y = -3(1-4)(1+2)

y = -3(-3)(3)

y = 27

Optimal Value = 27


Now we know the vertex is (1,27) and the x intercepts and now we just have to graph it.

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3.5 Graphing from Factored Form

Standard Form y = ax^2 + bx + c

Quadratic Formula

The quadratic formula ( x = -b + √b^2 - 4ac / 2a)is used to find the x intercepts of the standard form equation.


Example:

= x^2 + 3x – 4

x = -3 + √3^2 -4 (1)(-4)

= -3+ √9+16 / 2

= -3+√25 /2

= -3+5 /2

= 2/2

=1

=-3-5

=-8/2

=-4


X intercepts are -4 and 1.

Completing the square

y = 2x^2 +8x + 5


To complete the square first you have to bracket the first 2 terms and leave the last term on the side.


y = (2x^2 +8x) + 5


Than you want to factor out the 2. Do not factor the 2.

y = 2 (x^2 +4x) + 5


After that you want to divide the b value by 2 and square it.


y = 2 (x^2 +4x/2) + 5

y = 2 (x^2 +2x^2) + 5

y = 2 (x^2 +4x+4-4) + 5


Now you want to bring the -4 out of the bracket and multiply it with the a value. After multiplying it add it with 5.


y = 2 (x^2 +4x+4) -8+ 5

y = 2 (x^2 +4x+4) -3


Now you factor out the square in the bracket by square rooting everything inside of it.


y = 2 (x^2 +4x+4) -3

y = 2 (x+2) -3


And now you have completed the square and your equation is in vertex form.

Completing the square

Common Factoring

Common Factoring

Simple Trinomial

Simple Trinomials

Complex Trinomial

Complex trinomial

Perfect Squares

Perfect squares

Difference of Squares

Factoring,