A plane curve is graphed.A graph of the curve is in three sections separated by asymptotes at x equals negative pi and x = pi. The middle section of the graph passes through left-parenthesis negative start fraction pi over 2 end fraction comma 1 right-parenthesis, left-parenthesis 0 comma 2 right parenthesis and left-parenthesis start fraction pi over 2 end fraction comma 3 right-parenthesis.

Question

Question
Use the graph to answer the question.
What is the equation of the graphed trigonometric function?
(1 point)
Responses

y equals tangent left parenthesis one half x right parenthesis plus 2
Image with alt text: y equals tangent left parenthesis one half x right parenthesis plus 2

y equals tangent left parenthesis one half x right parenthesis minus 2
Image with alt text: y equals tangent left parenthesis one half x right parenthesis minus 2

y equals tangent left parenthesis 2 x right parenthesis minus 2
Image with alt text: y equals tangent left parenthesis 2 x right parenthesis minus 2

y equals tangent left-parenthesis 2 x right-parenthesis plus 2
Image with alt text: y equals tangent left-parenthesis 2 x right-parenthesis plus 2
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Answer 1

To determine the equation of the trigonometric function based on the provided information, we need to analyze the characteristics of the tangent function and the data points given.

1. Asymptotes: The asymptotes are at \( x = -\pi \) and \( x = \pi \), which suggest that the period of the tangent function is \( 2\pi \). However, since the transformation may compress or stretch the function, we need to consider the coefficient of \( x \) in the tangent function.

2. Points Given: The middle section of the graph passes through:

   • \( \left(-\frac{\pi}{2}, 1\right) \)
   • \( (0, 2) \)
   • \( \left(\frac{\pi}{2}, 3\right) \)

   The tangent function has a vertical asymptote at \( x = 0 \) (for standard tangent). However, since we see asymptotes at \( -\pi \) and \( \pi \), the graph could likely be compressed.

3. Vertical Translation: The points suggest that the function might have a vertical shift, as \( y \) ranges from below to above 2. The point at \( (0, 2) \) suggests a horizontal shift.

4. Periodicity: The tangent function has a natural period of \( \pi \) in its base form, but the presence of the vertical translations and the inputs provided at \( \frac{\pi}{2} \) and \( -\frac{\pi}{2} \) suggests a manipulation of the period.

5. Testing for Coefficients:

   • If we assume a transformation of \( \tan(kx) \) for some \( k \):
     • Since the asymptotes appear at \( -\pi \) and \( \pi \) when \( k = \frac{1}{2} \) (the period will be \( \frac{\pi}{k} = 2\pi \)).
   • Testing \( k = 2 \) would mean the period shrinks, and the asymptotes would appear every \( \frac{\pi}{2} \), which is not matching our asymptotes.

Based on the details, a shifted tangent function is more likely:

Given the options:

• The vertical shifts suggest we add or subtract a constant. The information hints that we have a \( +2 \) shift.

Thus:

• We infer that the correct equation is:

\[ y = \tan\left(\frac{1}{2} x\right) + 2 \]

This option correlates best with the data and shifts, confirming that the response is:

y equals tangent left parenthesis one half x right parenthesis plus 2