\[ y=2 \tan \left(\frac{1}{2} x+\frac{\pi}{4}\right)-1 \] Drag the movable black point to shift the function, the red points to set the vertical asymptotes, and the blue point at the correct set of coordinates. You may click on a point to verify its coordinates. Note that only one period of the function is shown.
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The function \( y=2 \tan \left(\frac{1}{2} x+\frac{\pi}{4}\right)-1 \) gives a fascinating example of how transformations affect the basic tangent curve. The coefficient of \(2\) indicates a vertical stretch, while the \(\frac{1}{2}\) inside the tangent function shows a horizontal stretch, making the period of the function \(2\pi\). Remember to visualize how the function shifts; by adjusting the x-values, you're essentially playing with the phase of the wave! Understanding and identifying vertical asymptotes is crucial when working with tangent functions. For this function, vertical asymptotes occur where the tangent function approaches infinity, specifically at points where \(\frac{1}{2}x + \frac{\pi}{4} = \frac{\pi}{2} + n\pi\) for any integer \(n\). This will help you accurately plot the behavior of the function and determine where it crosses the axes for a complete sketch!
