Find the equation of a sine function that has a vertical displacement 2 units down, a horizontal phase shift 60° to the right, a period of 30°, reflection in the y–axis and the amplitude of 3.
period of 30°: sin(12x)
amplitude of 3: 3sin(12x)
2 units down: 3sin(12x)-2
Now, things depend on whether the reflection is done before or after the shift.
Before: 3sin(12(-x-60°))-2
After: 3sin(-12(x-60°))-2
I guess do the reflection later and the down 2 last by multiplying by -1
60 deg = pi/3 rad, 30 deg = pi/6 rad
but I will do your degrees thing
3 sin (k x - 60)
when x = 30 , kx is 360 so k = 12
3 sin (12 x - 60)
reflect over y axis ?? strange but x ---> - x
3 sin (-12 x - 60)
move down 2
y = -2 + 3 sin (-12 x -60)
but sin -z = - sin z
y = -2 - sin (12 x + 60)
This all depends on the order in which you do the commands, really just making ll kinds of assumptions.
To find the equation of a sine function with the given characteristics, we can use the following general equation:
y = A * sin(B(x - C)) + D
where:
A is the amplitude
B is the coefficient of the horizontal shrink or stretch, which is determined by the period
C is the horizontal phase shift
D is the vertical displacement
From the given information, we can identify the values for each parameter as follows:
A = 3 (amplitude)
B = 360°/30° = 12 (period)
C = -60° (horizontal phase shift 60° to the right)
D = -2 (vertical displacement of 2 units down)
So, substituting these values into the general equation, we get:
y = 3 * sin(12(x - (-60))) - 2
Simplifying further:
y = 3 * sin(12x + 720) - 2
Finally, to account for the reflection in the y-axis, we multiply the equation by -1, resulting in:
y = -3 * sin(12x + 720) - 2
Therefore, the equation of the sine function satisfying the given characteristics is:
y = -3 * sin(12x + 720) - 2
To find the equation of a sine function with the given transformations, we will use the general form of the sine function:
y = A*sin(B(x - C)) + D
where:
A is the amplitude
B determines the period
C represents the horizontal phase shift
D is the vertical displacement
Given:
Amplitude (A) = 3
Period (360° / B) = 30° => B = 360° / 30° = 12
Horizontal phase shift (C) = 60° to the right => C = -60°
Vertical displacement (D) = 2 units down => D = -2
We also have the additional transformation of reflection in the y-axis, so we add a negative sign in front of our equation.
Therefore, the equation of the sine function is:
y = -3*sin(12(x + 60)) - 2