UNTITLED

Let's start with solving for: SIN² ø + COS²ø + 2COSø = 0

Think about the equation for the unit circle: x² + y² = 1

COS²ø + SIN²ø = 1, and it just happens that COS²ø and SIN²ø are in the equation.
So ...

SIN² ø + COS²ø + 2COSø = 0 can now be written as

1 + 2COSø = 0
COSø = - 1/2

*SOLVE FOR ALL REAL NUMBERS and your answers are:
2pi/3 + 2Kpi, K E I and 4pi/3 + 2Kpi, K E I




SKETCHING GRAPHS WITHOUT A CALCULATOR

EX. 1 Sketch f(x) = x (x-4)

* find where y = 0

x - 4 = 0
x = 4

x value is 4 so the axis of symmetry = 2

* substitute the value of the axis of symmetry for the x's in the equation

y = 2 (2 - 4)
y = 4 - 8
y = -4

vertex = (2, -4)

EX. 2 Sketch f(2x) = 2x (2x - 4)

* find where y = 0
b > 1
2x - 4 = 0
2x = 4
x = 2

x value is 2 so the axis of symmetry = 1

* substitute the value of the axis of symmetry for the x's in the equation

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

vertex = (1, -4)

* we horizontally compressed the original graph.


EX. 3 Sketch y f(x) = 1/2x (1/2x - 4)

* find where y = 0
0
1/2x - 4 = 0
1/2 x = 4
x = 8

x value is 8 so the axis of symmetry = 4

* substitute the value of the axis of symmetry for the x's in the equation

y = 1/2 (4) (1/2(4) - 4)
y = -4

_ _ _ _ _

SO ...

EX. 1 Start with y = x²

* Now graph y = 2f(x) by referring to the y = x² graph as a guideline.

The parabola should stretch:

* Now graph y = 2f (x) + 3 still referring to the y = x² graph as a guideline.

The parabola not only stretches, but also moves 3 units up due to the + 3 in the equation.

EX.2 Start with y = x²

* Now graph y = f (2x)²

This will make the parabola a compressed horizontal:

* Now graph y = f(2(x -8))²

This will take the original parabola 8 units to the right:

* Now graph y = f(2(x -8))² + 3

This will move the parabola 3 units upward.

* Sketch the graph of y = 1/2 (x + 4)² - 5

start with the y = x² graph to help you.

1. Because of the 1/2 the parabola will vertically compress.
2. Because of the 4 the parabols will move 4 units left.
3. Because of the - 5 the parabola will move 5 unit down.

For the next blog, I choose m0nkeh ! (I don't know who that is).