10 x2 a Write the function as a fraction involving sin andor

10. /(x)-2 a) Write the function as a fraction involving sin and/or cos. 3 6) b) Show your work in finding the equations of the vertical asymptotes. c) The function is graphed below. By substituting different integer values of k, find the equation of each asymptote you draw. Sketch the vertical asymptotes on the graph, and write the correct equation under each asymptote. d) Show your work in finding the a-intercepts of the function. If there are no r-intercepts, explain why e) By substituting different values of k, determine the location of each r-intercept on the graph below (if any exist). Then label each x-intercept on the graph with the coordinates (x,y). f) What is the range of the function? -3 -2 -3

Solution

a) f(x) = 2 csc ( pi/3 x+ pi/6 )

writing as a fraction

csc x = 1/ sin x

f(x) = 2 / sin ( pi/3 x + pi/6 )

b) setting denomintaor to 0 to find vertical asymptotes

sin ( pi/3 x + pi/6 ) = 0

x = pi/2 + 6k , 11/2 + 6k

c) substitution different values of k we get

vertical asymptotes is

x = -7/2 , -1/2 , 5/2

d) setting the function equal to 0 to find x intercepts

2 csc ( pi/3 x+ pi/6 ) = 0

csc ( pi/3 x+ pi/6 ) = 0

the equation has no x intercepts

since , csc ( pi/3 x+ pi/6 ) = 0 has no solutions

e) no x intercepts

f) range of the function is

(-infinity , -2 ] U [ 2 , infinity )