Question

I need help with this practice I’m struggling to solve it, I have already attempted it but I’m dealing uneasy on my final answer

Answer 1

y = 7 csc ( x /2) - 2

Step-by-step 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

`y=7\csc x`

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

`y=7\csc ((x)/(2))`

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

`(-9+5)/2`
`=-2`

which is the vertical shift.

Therefore, the final form of our function is

`\boxed{y=7\csc ((x)/(2))-2}`

which is our answer!