how do you graph y=-(1/4)sin(3/4x+pi/8? include the amplitude, period, vertical translation, horizontal shift, and table of values please.
The amplitude is 1/4.
Since the variable is x and not time, they should ask for the wavelength L, not the period.
2 pi/L = 3/4
L = (8/3)*pi
There is a horizontal shift of -pi/8 radians
We don't do plots. I suggest you make your own table of values and draw the graph. It will help you understand how the answers come about.
To graph the equation y=-(1/4)sin(3/4x+pi/8), we can break down the given equation to determine the values of amplitude, period, vertical translation, and horizontal shift.
1. Amplitude:
The amplitude of a sine function is the absolute value of the coefficient in front of the sine function. In this case, the amplitude is |-1/4|, which gives us an amplitude of 1/4.
2. Period:
The period of a sine function is calculated by dividing 2π by the coefficient of x. In this case, the coefficient is 3/4, so the period is (2π)/(3/4). Simplifying this expression, we get 8π/3.
3. Vertical Translation:
The vertical translation occurs due to any constant value that is added or subtracted to the original sine function. In our equation, there is no constant term added or subtracted, thus the vertical translation is 0.
4. Horizontal Shift:
The equation y=-(1/4)sin(3/4x+pi/8) contains an additional term inside the sine function, 3/4x+π/8. In this case, to determine the horizontal shift, we set the argument of the sine function (3/4x+π/8) equal to zero and solve for x:
3/4x + π/8 = 0
3/4x = -π/8
x = -(π/8)(4/3)
x = -π/6
Therefore, the horizontal shift is π/6 units to the right.
5. Table of Values:
To create a table of values, choose various x-values within your desired range and plug them into the equation to find the corresponding y-values. For simplicity, we will use increments of π/6 for x.
x | y=-(1/4)sin(3/4x+π/8)
----------------------------------------------------
-π | -(1/4)sin(-π/2+π/8) ≈ 0.191
-π/6 | -(1/4)sin(-π/4+π/8) ≈ 0.354
0 | -(1/4)sin(0+π/8) ≈ -0.25
π/6 | -(1/4)sin(π/4+π/8) ≈ -0.354
π | -(1/4)sin(π/2+π/8) ≈ -0.191
Now, with the amplitude, period, vertical translation, horizontal shift, and table of values, you have all the information needed to graph the equation y=-(1/4)sin(3/4x+π/8).