State the amplitude, period, phase shift, and vertical shift of the function y= -3 (1/2x- 2/3 pi)-4
I have a feeling that you have a typo and this is supposed to be either a sine or cosine function
Assuming I am correct .....
for any sine function in the form:
y = a sin (k(Ø + d)) + c
|a| is the amplitude
the period is 2π/k
the phase shift is d units to the left, or -d units to the right
the vertical shift is c
apply to your equation.
the only tricky part I see is in the period
change your equation to
y = -3sin((1/2)(x - 4/3) ) - 4
so the phase shift is 4/3 units to the right.
To find the amplitude, period, phase shift, and vertical shift of the function y = -3(1/2x - 2/3π) - 4, let's break it down step by step:
1. Amplitude: In this case, the amplitude can be determined by looking at the coefficient of x, which is 1/2. The absolute value of this coefficient is 1/2, so the amplitude is 1/2.
2. Period: The period of a trigonometric function is usually determined by dividing 2π by the coefficient of x. In this case, the coefficient of x is 1/2. So, to find the period, we can use the formula:
Period = 2π / (coefficient of x)
= 2π / (1/2)
= 4π
Therefore, the period of this function is 4π.
3. Phase Shift: The phase shift of a function is determined by solving the equation inside the parentheses for x that makes it equal to zero. In this case, the equation 1/2x - 2/3π = 0 can be solved as follows:
1/2x - 2/3π = 0
1/2x = 2/3π
x = (2/3π) / (1/2)
x = 4/3π
So, the phase shift for this function is 4/3π.
4. Vertical Shift: The vertical shift represents the vertical displacement of the graph from the x-axis. In this case, the constant term -4 in the function y = -3(1/2x - 2/3π) - 4 is the vertical shift. Since it is -4, the graph is shifted downwards by 4 units.
To summarize:
- The amplitude is 1/2.
- The period is 4π.
- The phase shift is 4/3π.
- The vertical shift is -4.