The graph \(y = a \tan (bx -c) + d\) has the following characteristics.

• The amplitude is not defined
• The period is given by \(\displaystyle \frac{ \pi}{b}\).
• Two consecutive vertical asymptotes can be found by solving the equations \(bx-c = -\frac{\pi}{2}\) and \(bx-c = \frac{\pi}{2}\)
• The domain excludes all values of \(x\) which are solutions to \(\cos(bx-c)=0.\) That is, \(D = \lbrace x \in \mathbb{R} \; | \; \cos(bx-c) \not = 0 \rbrace\)

The graph \(y = a \cot (bx -c) + d\) has the following characteristics.

• The amplitude is not defined
• The period is given by \(\displaystyle \frac{ \pi}{b}\).
• Two consecutive vertical asymptotes can be found by solving the equations \(bx-c = 0\) and \(bx-c = \pi\)
• The domain excludes all values of \(x\) which are solutions to \(\sin(bx-c)=0.\) That is, \(D = \lbrace x \in \mathbb{R} \; | \; \sin(bx-c) \not = 0 \rbrace\)

The graph \(y = a \sec (bx -c) + d\) has the following characteristics.

• The amplitude is not defined
• The period is given by \(\displaystyle \frac{2 \pi}{b}\).
• Two consecutive vertical asymptotes can be found by solving the equations \(bx-c = -\frac{\pi}{2}\) and \(bx-c = \frac{\pi}{2}\)
• The domain excludes all values of \(x\) which are solutions to \(\cos(bx-c)=0.\) That is, \(D = \lbrace x \in \mathbb{R} \; | \; \cos(bx-c) \not = 0 \rbrace\)

The graph \(y = a \csc (bx -c) + d\) has the following characteristics.

• The amplitude is not defined
• The period is given by \(\displaystyle \frac{2 \pi}{b}\).
• Two consecutive vertical asymptotes can be found by solving the equations \(bx-c = 0\) and \(bx-c = 2 \pi\)
• The domain excludes all values of \(x\) which are solutions to \(\sin(bx-c)=0.\) That is, \(D = \lbrace x \in \mathbb{R} \; | \; \sin(bx-c) \not = 0 \rbrace\)