given the function y=2sin3x-1, identify the following below. also show its graph for the primary cycle. (identifying parts of a trig. equation)

asymptotes=
k=
c=
axis=
primary cycle=
period=

Get you a ti-83 and graph it...should show you all of this

To identify the different parts of the given trigonometric equation, y = 2sin(3x) - 1, and draw its graph for the primary cycle, let's break it down step by step:

1. Asymptotes:
Trigonometric functions, such as sine, do not have asymptotes like rational functions. Therefore, there are no asymptotes in this equation.

2. k (Vertical Shift):
The value of k determines the vertical shift of the graph. In this equation, the constant term -1 represents the vertical shift. So, k = -1.

3. c (Phase Shift):
The phase shift represents the horizontal shift of the graph in radians. In this equation, since there is no extra term involving x, there is no phase shift. So, c = 0.

4. Axis:
The axis of symmetry is a vertical line that cuts the graph into two identical halves. For a sine function, the axis of symmetry is the midline, which is the average between the maximum and minimum values of the function. In this case, the maximum value of the function is 2 and the minimum value is -1. Therefore, the axis of symmetry is at y = (2 - 1) / 2 = 0.5.

5. Primary Cycle:
To graph the primary cycle, we need to identify important points on the graph:
a. Maximum and Minimum values: The maximum value of the function is 2, while the minimum value is -1.
b. x-intercepts: To find the x-intercepts, set y = 0 and solve for x:
2sin(3x) - 1 = 0
2sin(3x) = 1
sin(3x) = 1/2
3x = π/6 or 5π/6 (by using the unit circle)
x = π/18 or 5π/18

6. Period:
The period of a sine function is the horizontal length of one complete cycle. The formula to calculate the period is: T = 2π/b, where b represents the coefficient of x. In this case, b = 3, so the period is T = 2π/3.

Now, let's plot the graph for the primary cycle, considering the information we gathered:
- Start by drawing the x-axis and y-axis.
- Mark the y-intercept at (0, k) = (0, -1).
- Draw the maximum and minimum points at (π/18, 2) and (5π/18, -1), respectively.
- Draw the x-intercepts at (π/18, 0) and (5π/18, 0).
- Sketch the curve smoothly connecting these points, remembering it has a period of 2π/3.
-Finally, label the axis of symmetry at y = 0.5.

There you go! You have identified the different parts of the trigonometric equation and graphed its primary cycle.