Normally, the first 3 key points of a sine curve are (0,0), (90,1), and (180,0). If the function is changed to y=-2sin(3x-120)+5, the key points given above change to...

Please explain this to me so that I can do it myself.

Interpretation #1: you want to keep those same x values

sub in the x values of the original points:
(0,0) ---> y = 2sin(-120) + 5 = 2(√3/2) + 5 = √3 + 5
(0,0) ----> (0, √3+5)

for (90,1)
y = 2sin(150) + 5 = 2(1/2) + 5 = 6
(90,1) ----( 90,6)

you do the last one.

Interpretation #2: You are looking for the 3 points that describe the start value, the max value, then back to the start value, (half a period)

change the equation to y = 2 sin 3(x - 40) + 5
that tells me that the curve y = 2 sin 3x has been moved horizontally 40° to the right and up 5
so the period is 360°/3 = 120°
So I would make my "key point" values of x equal to
40, 70, 100
(notice that a complete period would be shown using the x values of 40, 70, 100, 130)
let x=40, y = 2 sin 3(40-40) + 5 = 0+5 = 5
(0,0) --- (40,5)
let x = 70 , y = 2 sin 3(70-40) + 5 = 2(1) + 5 = 7
(90,1) ----> (70,7)

you try the last one

To find the key points of the modified sine curve, y = -2sin(3x-120) + 5, we can follow these steps:

Step 1: Identify the amplitude: The amplitude of a sine curve is the distance between the highest and lowest points of the curve. In this case, the coefficient of sin is -2. Since the amplitude is always positive, we take the absolute value, which is 2.

Step 2: Determine the period: The period of a sine curve is the length of one complete cycle. In a regular sine curve, the period is 360 degrees or 2π radians. However, in this case, we have sin(3x - 120), which means the period is shortened. To find the new period, we divide the regular period (360 degrees or 2π radians) by the coefficient of x, which is 3. Therefore, the new period is 120 degrees or 2π/3 radians.

Step 3: Find the horizontal shift: The horizontal shift determines the starting point of the sine curve. In this case, we have sin(3x - 120), which means the curve is shifted 120 degrees to the right. This shift is in the opposite direction of what it appears, so we consider it as 120 degrees to the right.

Now, let's apply these steps to find the new key points:

Key Point 1: (0,0) - Since the starting point of the curve is shifted to the right by 120 degrees, the new x-coordinate will be 120 degrees. To find the y-coordinate, substitute x = 120 into the equation: y = -2sin(3(120) - 120) + 5.

Key Point 2: (90,1) - Similarly, the new x-coordinate will be 90 + 120 = 210 degrees. Substitute this into the equation to find the y-coordinate: y = -2sin(3(210) - 120) + 5.

Key Point 3: (180,0) - Once again, add the horizontal shift of 120 degrees to the original x-coordinate of 180 degrees to get 300 degrees. Plug this into the equation to find the y-coordinate: y = -2sin(3(300) - 120) + 5.

Using these steps, you can calculate the new key points based on the modified sine curve, y = -2sin(3x-120) + 5.