How do you graph #y=5csc3(x-pi)-5#?

Question

Quinn Anderson · Accepted Answer

As below.

#y = 5 csc (3x - 3pi) - 5# Standard form of cosecant function is #y = A csc(Bx - C) + D# #A = 5, B = 3, C = 3 pi, D = -5# #Amplitude = |A| = "NONE for cosecant function"# #"Period " = (2pi) / |B| = (2pi) / 3 # #"Phase Shift " = -C / B = (3pi) / 3 = pi, " " pi " to the RIGHT"# #"Vertical Shift " = D = -5# graph{5 csc(3x - 3 pi) - 5 [-10, 10, -5, 5]}

Adam Aguirre · Answer

undefined To graph $y = 5\csc(3(x - \pi)) - 5$, follow these steps:

1. Identify the vertical asymptotes: Set the expression inside the cosecant function equal to zero and solve for $x$.
$$3(x - \pi) = 0 $$
$$x - \pi = 0 $$
$$x = \pi $$

2. Plot the vertical asymptote at $x = \pi$.

3. Determine the amplitude of the graph, which is the absolute value of the coefficient of the cosecant function, $5$.

4. Graph the cosecant function: Start by plotting key points of the cosecant function, which occur at the zeros of the sine function. For $y = \csc(x)$, these occur at multiples of $\pi$.
$$x_1 = \pi - \pi = 0 $$
$$x_2 = \pi + \pi = 2\pi $$
$$x_3 = \pi + 2\pi = 3\pi $$
$$x_4 = \pi - 2\pi = -\pi $$
$$x_5 = \pi - 3\pi = -2\pi $$

5. Plot these points and draw the curve. Remember that the graph approaches the asymptotes but never crosses them.

6. Multiply the graph of the cosecant function by the amplitude, $5$, and then subtract $5$ to shift the graph downward.

7. Label the graph as necessary.

8. Your final graph should resemble the general shape of the cosecant function, but it will be vertically stretched by a factor of $5$ and shifted downward by $5$.