Question

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which function is shown on the graph?
\\( f(x) = -\frac{1}{2}cos x \\)
\\( f(x) = -\frac{1}{2}sin x \\)
\\( f(x) = \frac{1}{2}sin x \\)
\\( f(x) = \frac{1}{2}cos x \\)
graph: a trigonometric graph with x-axis marked at -π/2, 0, π/2, π, 3π/2, 2π, y-axis marked at -2, -1, 0, 1, 2. the graph starts from the left, crosses the x-axis at -π/2, reaches a peak at x=0, crosses the x-axis at π/2, has a trough at x=π, crosses the x-axis at 3π/2, and then rises slightly towards 2π.

Explanation:

1. First, recall the properties of sine and cosine functions:

• The cosine function \( y = \cos x \) has a maximum at \( x = 0 \), while the sine function \( y=\sin x \) has a value of \( 0 \) at \( x = 0 \). Looking at the graph, at \( x = 0 \), the function has a maximum (or a non - zero value), so it is a cosine - type function, eliminating the sine function options (\(f(x)=-\frac{1}{2}\sin x\) and \(f(x)=\frac{1}{2}\sin x\)).

1. Then, analyze the sign of the cosine function:

• The standard cosine function \( y = \cos x \) has a maximum at \( x = 0 \). The given function at \( x = 0 \) has a positive value. The function \( y = A\cos x \): if \( A>0 \), the graph has the same general shape as \( y = \cos x \) (maximum at \( x = 0 \)) with amplitude \(|A|\). The amplitude of the function here is \( \frac{1}{2} \) (since the maximum value seems to be \( \frac{1}{2} \) from the graph's scale). The function \( f(x)=-\frac{1}{2}\cos x \) would be a reflection of \( y = \frac{1}{2}\cos x \) over the \( x \) - axis, which would have a minimum at \( x = 0 \), but our graph has a maximum at \( x = 0 \). So the function is \( f(x)=\frac{1}{2}\cos x \).

Answer:

\( f(x)=\frac{1}{2}\cos x \)