Question

what is the equation of the reciprocal function?
$y = -\frac{1}{2}\csc\frac{x}{4}$
(type an expression using x as the variable. use integers or fractions for any numbers in the expression.)
choose the correct graph below.
○ a.
○ b.
○ c.
○ d.

Explanation:

Step1: Recall the reciprocal of sine

The cosecant function \( \csc x \) is the reciprocal of the sine function, so \( \csc x=\frac{1}{\sin x} \). For the function \( y = -\frac{1}{2}\csc\frac{x}{4} \), we can rewrite it using the reciprocal identity: \( y = -\frac{1}{2}\cdot\frac{1}{\sin\frac{x}{4}} \), which simplifies to \( y = -\frac{1}{2\sin\frac{x}{4}} \).

Step2: Analyze the graph (optional for the equation part, but to confirm)

The reciprocal function of a sine function will have vertical asymptotes where the sine function is zero. The function \( \sin\frac{x}{4}=0 \) when \( \frac{x}{4}=n\pi \) ( \( n \) is an integer), so \( x = 4n\pi \). The negative sign and the coefficient \( -\frac{1}{2} \) will reflect the graph over the x - axis and vertically compress it. But for the equation part, using the reciprocal identity of cosecant and sine is sufficient.

1. First, recall the properties of \( y = -\frac{1}{2}\csc\frac{x}{4} \). The cosecant function \( \csc t=\frac{1}{\sin t} \), here \( t = \frac{x}{4} \). The sine function \( \sin\frac{x}{4} \) has a period of \( T=\frac{2\pi}{\frac{1}{4}} = 8\pi \), so the cosecant function also has a period of \( 8\pi \).
2. The negative sign in \( y = -\frac{1}{2}\csc\frac{x}{4} \) reflects the graph of \( y=\frac{1}{2}\csc\frac{x}{4} \) over the x - axis. When \( \sin\frac{x}{4}>0 \), \( \csc\frac{x}{4}>0 \), and with the negative sign, \( y = -\frac{1}{2}\csc\frac{x}{4}<0 \). When \( \sin\frac{x}{4}<0 \), \( \csc\frac{x}{4}<0 \), and with the negative sign, \( y = -\frac{1}{2}\csc\frac{x}{4}>0 \).
3. Let's analyze the options:

• Option A: The graph has positive - valued cosecant - like curves in the regions where we would expect negative values (based on the negative sign) and vice - versa. So A is incorrect.
• Option B: Let's check the sign of the function. For \( y = -\frac{1}{2}\csc\frac{x}{4} \), when \( x = 0 \), \( \sin(0)=0 \) (asymptote), but near \( x = 2\pi \), \( \sin\frac{2\pi}{4}=\sin\frac{\pi}{2}=1>0 \), so \( y = -\frac{1}{2}\csc\frac{2\pi}{4}=-\frac{1}{2}\cdot1 = -\frac{1}{2}<0 \). In graph B, the curve near \( x = 2\pi \) is positive, so B is incorrect.
• Option C: When \( \sin\frac{x}{4}>0 \), \( y = -\frac{1}{2}\csc\frac{x}{4}<0 \) and when \( \sin\frac{x}{4}<0 \), \( y = -\frac{1}{2}\csc\frac{x}{4}>0 \). Also, the period is \( 8\pi \), and the vertical asymptotes are at \( x = 4n\pi \). The shape and sign of the graph in option C match the properties of \( y = -\frac{1}{2}\csc\frac{x}{4} \).
• Option D: The graph has positive - valued cosecant - like curves in the regions where we would expect negative values (based on the negative sign) and vice - versa. So D is incorrect.

Answer:

\( y = -\frac{1}{2\sin\frac{x}{4}} \) (or \( y = -\frac{1}{2}\csc\frac{x}{4} \) as originally given, but the form using sine is also a valid expression for the reciprocal function)

For the graph part (choosing the correct graph):