Mathematics » Functions II » The Tangent Function

# Revision of The Tangent Function

## 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 $$\theta =\text{270}\text{°}$$

### 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.

 $$a<0$$ $$a>0$$ $$q>0$$ $$q=0$$ $$q<0$$