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

$csc(x) = \frac{1}{sin(x)}$

f(x) = A csc⁡(ωx −ϕ)+B

A = Amplitude = |A|

$Period = \frac{2\pi}{\omega}$

$Phase\ Shift = \frac{\phi}{\omega}$

Y-Shift = B

Asymptotes of csc

$x = \frac{k}{\pi}$

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:

$Period = \frac{2\pi}{\omega}$

$Period = \frac{2\pi}{1}$

$Period = 2\pi$

Find the Phase Shift

$Phase\ Shift = \frac{\phi}{\omega}$

$Phase\ Shift = \frac{\frac{\pi}{4}}{1}$

$Phase\ Shift = \frac{\pi}{4}$

Find the Asymptotes

We know $csc(x) = \frac{1}{sin(x)}$ and when the denominator of $\frac{1}{sin(x)}$ 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.

$Phase\ Shift = \frac{\pi}{4}$

$x = (k\pi) + (\frac{\pi}{4})$

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

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

Answer:

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

Step-by-step explanation:

Correct on edg assignment