## 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 \(0<a<1\), branches of \(f(\theta)\) are less steep and curve more.

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.

\(a<0\)

\(a>0\)

\(q>0\)

\(q=0\)

\(q<0\)

**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 \(0<a<1\), branches of \(f(\theta)\) are less steep and curve more.

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.

\(a<0\) | \(a>0\) | |

\(q>0\) | ||

\(q=0\) | ||

\(q<0\) |