Question

given the equation ( y = 6 cscleft( \frac{5pi}{4}x+\frac{35pi}{4}
ight)
the period is:
the horizontal shift is: units to the select an answer
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Explanation:

Step1: Recall the formula for the period of \(y = A\csc(Bx - C)+D\)

The period of \(y = A\csc(Bx - C)+D\) is \(\frac{2\pi}{|B|}\).
For \(y = 6\csc(\frac{5\pi}{4}x+\frac{35\pi}{4})\), \(B=\frac{5\pi}{4}\).
So the period \(T=\frac{2\pi}{\frac{5\pi}{4}}\)

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Step2: Recall the formula for the horizontal shift of \(y = A\csc(Bx - C)+D\)

The horizontal shift is \(\frac{C}{B}\).
Rewrite \(y = 6\csc(\frac{5\pi}{4}x+\frac{35\pi}{4})\) as \(y = 6\csc(\frac{5\pi}{4}(x + 7))\), so \(B=\frac{5\pi}{4}\) and \(C=- \frac{35\pi}{4}\)
The horizontal shift \(h=\frac{-\frac{35\pi}{4}}{\frac{5\pi}{4}}\)

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A negative shift means 7 units to the left.

Answer:

The period is: \(\frac{8}{5}\)
The horizontal shift is: \(7\) units to the left.