Question

The graph of a trigonometric function is shown. The graph has asymptotes at integer multiples of pi and crosses the x-axis at integer multiples of start fraction pi over 2 end fraction. Each cycle of the graph decreases from y equals infinity to y equals negative infinity.
Question
Use the graph to answer the question.
Which function is shown in the graph?
(1 point)
Responses

f left parenthesis x right parenthesis equals tangent x
Image with alt text: f left parenthesis x right parenthesis equals tangent x

f left parenthesis x right parenthesis equals secant x
Image with alt text: f left parenthesis x right parenthesis equals secant x

f left parenthesis x right parenthesis equals cosecant x
Image with alt text: f left parenthesis x right parenthesis equals cosecant x

f left parenthesis x right parenthesis equals cotangent x
Image with alt text: f left parenthesis x right parenthesis equals cotangent x

Answer 1

Based on your description, the graph has vertical asymptotes at integer multiples of \(\pi\) and crosses the x-axis at integer multiples of \(\frac{\pi}{2}\). Additionally, it decreases from \(y = \infty\) to \(y = -\infty\) in each cycle.

This behavior is characteristic of the tangent function, \(f(x) = \tan(x)\). The tangent function has:

- Vertical asymptotes at \(x = \frac{\pi}{2} + n\pi\) (for integer \(n\)), which correspond to the points you noted as appearing at integer multiples of \(\pi\) when considering where the function is undefined.
- It crosses the x-axis at \(x = n\pi\) (for integer \(n\)), where its value is zero.

Therefore, the function shown in the graph is:
**\(f(x) = \tan(x)\)**.