Questions: Graph two periods of the given tangent function. y=6 tan (x/5)

Solution Steps

The general form of the tangent function is \( y = a \tan(bx) \). The period of the tangent function \( \tan(bx) \) is given by \( \frac{\pi}{b} \).

For the given function \( y = 6 \tan\left(\frac{x}{5}\right) \):

• \( b = \frac{1}{5} \)
• The period \( T \) is \( \frac{\pi}{\frac{1}{5}} = 5\pi \)

The tangent function has vertical asymptotes where the function is undefined. For \( y = 6 \tan\left(\frac{x}{5}\right) \), the vertical asymptotes occur at: \[ x = \frac{5\pi}{2} + k \cdot 5\pi \] where \( k \) is an integer.

To graph two periods of \( y = 6 \tan\left(\frac{x}{5}\right) \):

• The function will repeat every \( 5\pi \).
• The graph should show the function from \( x = -5\pi \) to \( x = 5\pi \).

Final Answer