Revision of The Tangent Function

Functions of the form \(y = \tan\theta\) for \(\text{0}\text{°} \leq \theta \leq \text{360}\text{°}\)

The dashed vertical lines are called the asymptotes. The asymptotes are at the values of θ where \(\tan\theta\) is not defined.

• Period: \(\text{180}\text{°}\)

• Domain: \(\{\theta : \text{0}\text{°} \le \theta \le \text{360}\text{°}, \theta \ne \text{90}\text{°}; \text{270}\text{°}\}\)

• Range: \(\{f(\theta):f(\theta)\in ℝ\}\)

• \(x\)-intercepts: \((\text{0}\text{°};0)\), \((\text{180}\text{°};0)\), \((\text{360}\text{°};0)\)

• \(y\)-intercept: \((\text{0}\text{°};0)\)

• Asymptotes: the lines \(\theta =\text{90}\text{°}\) and \(\theta =\text{270}\text{°}\)

Functions of the form \(y = a \tan \theta + q\)

Tangent functions of the general form \(y = a \tan \theta + q\), where \(a\) and \(q\) are constants.

The effects of \(a\) and \(q\) on \(f(\theta) = a \tan \theta + q\):

• The effect of \(q\) on vertical shift

  • For \(q>0\), \(f(\theta)\) is shifted vertically upwards by \(q\) units.

  • For \(q<0\), \(f(\theta)\) is shifted vertically downwards by \(q\) units.

• The effect of \(a\) on shape

  • For \(a>1\), branches of \(f(\theta)\) are steeper.

  • For \(0

  • For \(a<0\), there is a reflection about the \(x\)-axis.

  • For \(-1 < a < 0\), there is a reflection about the \(x\)-axis and the branches of the graph are less steep.

  • For \(a < -1\), there is a reflection about the \(x\)-axis and the branches of the graph are steeper.

	\(a<0\)	\(a>0\)
\(q>0\)
\(q=0\)
\(q<0\)