How do you graph #y=2csc[2(x+pi/6)]#?

graph{2 csc(2x + pi/3) [-10, 10, -5, 5]}

To graph the function y = 2csc[2(x + π/6)], follow these steps:

1. Identify the key features of the function:

   • The coefficient of x inside the function, which affects the period of the graph.
   • The phase shift, determined by the constant inside the function, which shifts the graph horizontally.
   • The vertical stretch factor, which multiplies the whole function vertically.

2. Period: The period of the function is determined by the coefficient of x inside the function. In this case, it's 2, which means the period is π/2.

3. Phase Shift: The phase shift is determined by the constant inside the function. In this case, it's -π/6. Since it's subtracted from x, the phase shift is to the left by π/6 units.

4. Vertical Stretch Factor: The vertical stretch factor is 2, which means the graph will be vertically stretched by a factor of 2.

5. Vertical Asymptotes: Since the function is csc, it will have vertical asymptotes where the sine function crosses the x-axis. These occur whenever sin[2(x + π/6)] = 0. Solve for x to find the vertical asymptotes.

6. Plot Points: Choose some key points to plot the graph. These points should include critical points such as the maximums, minimums, and points where the graph intersects the asymptotes.

7. Plot the Graph: Use the information gathered from the steps above to plot the graph accurately, showing the shape of the function over its period.

8. Repeat: Repeat the pattern of the function to extend the graph further if needed.

To graph (y = 2\csc[2(x+\frac{\pi}{6})]), you can start by identifying key features of the cosecant function and applying transformations.

The basic cosecant function has vertical asymptotes at (x = k\pi), where (k) is an integer, and its period is (2\pi). It has a minimum value of -1 and a maximum value of 1.

The given function (y = 2\csc[2(x+\frac{\pi}{6})]) has an amplitude of 2, which means the graph will oscillate between -2 and 2. It is also vertically stretched by a factor of 2 compared to the basic cosecant function.

The period of the function is (\frac{2\pi}{2} = \pi), which means the graph will complete one full cycle every (\pi) units.

The function is shifted horizontally by (-\frac{\pi}{6}) units, which means the graph will be shifted to the left by (\frac{\pi}{6}) units.

To graph the function, start by plotting the vertical asymptotes at (x = k\pi - \frac{\pi}{6}), where (k) is an integer. Then, mark the maximum and minimum points on each interval between the asymptotes, considering the amplitude of 2. Connect these points with smooth curves to complete the graph.

It's also helpful to plot the basic cosecant function (y = \csc(x)) as a reference, and then apply the transformations to sketch the graph of (y = 2\csc[2(x+\frac{\pi}{6})]).