Question

The graph of y=csc(x-pi/4)-3 is shown. What is the period of the function? Where are the asymptotes of the function? What is the range of the function? y_< ? y_>? Graph

Answer 1

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Prerequisites:

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You need to know

`csc(x) = (1)/(sin(x))`

f(x) = A csc⁡(ωx −ϕ)+B

A = Amplitude = |A|

`Period = (2\pi)/(\omega)`

`Phase\ Shift = (\phi)/(\omega)`

Y-Shift = B

Asymptotes of csc

`x = (k)/(\pi)`

k = any constant integer number like -3, -2, -1, 0, 1, 2, 3...

Range of csc x

The ranges is from
`(-\infty,-1]U[1, \infty)`

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Step By Step Explanation:

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Find the Period of the function:

`Period = (2\pi)/(\omega)`

`Period = (2\pi)/(1)`

`Period = 2\pi`

Find the Phase Shift

`Phase\ Shift = (\phi)/(\omega)`

`Phase\ Shift = ((\pi)/(4))/(1)`

`Phase\ Shift = (\pi)/(4)`

Find the Asymptotes

We know
`csc(x) = (1)/(sin(x))` and when the denominator of
`(1)/(sin(x))` is equal to 0, we have an asymptote. So sin = 0 at k*pi .

We know the asymptotes for
`csc(\theta)` is where
`x = k\pi`.

The function that is presented has asymptotes at
`x = k\pi`.+ Phase Shift.

`Phase\ Shift = (\pi)/(4)`

`x = (k\pi) + ((\pi)/(4))`

Now, just plugin an integer for k and you will find the asymptotes or you could say the asymptotes are at
`x = (k\pi) + ((\pi)/(4))`

Find the range.

We know the range for csc x =
`(-\infty,-1]U[1, \infty)`. Since we have a Y-Shift of -3, we have to adjust by subtracting -3 from 1 and -1.

1 - 3 = -2

-1 - 3 = -4

New range =
`(-\infty,-4]U[-2, \infty)`.

Answer 2

2pi, pi/4 + npi, -4, -2

Explanation:

Correct on edg assignment