Can someone help me with my algebra 2 homework ?

Question

You don't have to tell me the answers to all, I just don't have a graphics calculator, and need to know how to do it. thank you!

Find the amplitude and period of each.

1. y=3 sin feta
2. y= 2.5 cos feta
3. y= 6 sin 4feta
4. y= 2 tan feta
5. y= -2 cos 2feta
6. y= 1.5 sin feta

Identify phase shift & vertical translation of each, from its parent function.

( you can just tell me how to do it, but here is an example )

y= sin(feta + 90degrees) + 2


Last, I promise !

Describe transformation of each.

1. y= 3 sin 3feta-1

2. y= -cos(feta-45degrees) + 2

Answer 1

Before I start, the angle variable "feta" is really spelled "theta."

To calculate the amplitude of a sinusoidal function, look at the coefficient (the multiplier) outside of the function, if the function is sine or cosine.

For example, the amplitude of y = 3 sin (theta) is 3.

This is not the case with tangent, since it has a vertical asymptote at x = pi/2. So tangent has infinite amplitude.

To find the period, look at what is inside the trigonometric function (inside the parenthesis). Divide 2*pi by the coefficient inside the parenthesis, and you will get the period. Remember, if there is no number inside the parenthesis, then the coefficient is one.

Thus, the period for y = 3 sin (theta) is 2 pi (coefficient is 1). Also, the period for y = 6 sin (4*theta) is 0.5 pi (2 pi / 4).

To find the phase shift, you have to look inside the parenthesis again. If you are adding to the angle, you will shift to the left. If you are subtracting from the angle, you will shift the function right. So for y = sin(theta + 90 degrees) + 2, the sine function will be shifted to the left 90 degrees.

The vertical shift is found outside the trigonometric function. If you are adding to the function, you will move it up. If you are subtracting, you will move it down. This makes sense since you are adding or subtracting directly to the y value. For your example y = sin(theta + 90 degrees) + 2, you will move the sine function up 2 units.

When you describe a transformation, you will need to apply all of the above characteristics: amplitude, period, phase shift, and vertical translation. Here's the first example y = 3 sin (3*theta) - 1

Amplitude: Increases by factor of 3
Period: Shrinks by a factor of 3
Phase Shift: None
Vertical Translation: Down 1 unit

Things get really tricky when you have an equation like this:
y = 3 * (sin (3*(theta-180)) + 2)
Is the vertical translation up 2 units? Not so fast. Multiply out the 3 so the equation looks like this:
y = 3 * sin (3*(theta-180)) + 6
Now you can see the vertical displacement is really 6. You should do the same with the stuff inside the trigonometric function.

Answer 2

"feta" should be "theta," Θ, the "th" equivalent in the Greek alphabet.

1. y=3 sin feta -- period 360°, amplitude 3
2. y= 2.5 cos feta -- period 360°, amplitude 2.5
3. y= 6 sin 4feta -- period 360/4 = 90°, amp 6
4. y= 2 tan feta -- period of tan is 180°, vertically stretched by factor of 2
5. y= -2 cos 2feta -- period 360/2 = 180, amp 2 (and the - flips it upside down)
6. y= 1.5 sin feta -- per. 360, amp 1.5

1. y= 3 sin 3feta-1 -- period 360/3 = 120, amp 3, which is horiz. compression by factor of 3, vertical stretch by factor of 3.

2. y= -cos(feta-45degrees) + 2 -- phase shifted 45° right, flipped upside down, vert. shifted 2 up.