Answer:
y = 7 csc ( x /2) - 2
Explanation:
The parent csc x function has the range ( - ∞, -1] U [1,∞ ) infinity, meaning its "width" is 2. Now our function has range ( - ∞, -9] U [5,∞ ), with a width of 5 - (-9) = 14, which is two times as much as that of the parent function function. This tells us that the function is vertically stretched by a factor of 14 / 2 = 7, and so we have
[tex]y=7\csc x[/tex]
Now, the parent function csc x has vertical asympototes at x = 0 and x = π. Our function, however, has consecutive asymptotes at x = 0 and x = 2 π. Meaning our functin is streched horizontally by a factor of 2. Therefore, we do x --> x / 2 and so we have
[tex]y=7\csc (\frac{x}{2})[/tex]
Lastly, we observe that the function is not symmetric about the x-axis, whereas the parent function is. This tells us that some vertical shifting is involved. How much vertical shifting exactly?
We find the answer by finding the midpoint
[tex](-9+5)/2[/tex][tex]=-2[/tex]
which is the vertical shift.
Therefore, the final form of our function is
[tex]\boxed{y=7\csc (\frac{x}{2})-2}[/tex]
which is our answer!
