Mathematics » Functions II » The Tangent Function

# Sketching Tangent Graphs

## Example

### Question

Sketch the graph of $$f(\theta) = \tan \cfrac{1}{2}(\theta – \text{30}\text{°})$$ for $$-\text{180}\text{°} \leq \theta \leq \text{180}\text{°}$$.

### Examine the form of the equation

From the equation we see that $$0 < k < 1$$, therefore the branches of the graph will be less steep than the standard tangent graph $$y = \tan \theta$$. We also notice that $$p < 0$$ so the graph will be shifted to the right on the $$x$$-axis.

### Determine the period

The period for $$f(\theta) = \tan \cfrac{1}{2}(\theta – \text{30}\text{°})$$ is:

\begin{align*} \text{Period} &= \cfrac{\text{180}\text{°}}{|k|} \\ &= \dfrac{\text{180}\text{°}}{\cfrac{1}{2}} \\ &= \text{360}\text{°} \end{align*}

### Determine the asymptotes

The standard tangent graph, $$y = \tan \theta$$, for $$-\text{180}\text{°} \leq \theta \leq \text{180}\text{°}$$ is undefined at $$\theta = -\text{90}\text{°}$$ and $$\theta = \text{90}\text{°}$$. Therefore we can determine the asymptotes of $$f(\theta) = \tan \cfrac{1}{2}(\theta – \text{30}\text{°})$$:

• $$\cfrac{-\text{90}\text{°}}{\text{0.5}} + \text{30}\text{°} = -\text{150}\text{°}$$
• $$\cfrac{\text{90}\text{°}}{\text{0.5}} + \text{30}\text{°} = \text{210}\text{°}$$

The asymptote at $$\theta = \text{210}\text{°}$$ lies outside the required interval.

### Plot the points and join with a smooth curve

Period: $$\text{360}\text{°}$$

Domain: $$\{ \theta: -\text{180}\text{°} \leq \theta \leq \text{180}\text{°}, \theta \ne -\text{150}\text{°} \}$$

Range: $$(-\infty;\infty)$$

$$y$$-intercepts: $$(\text{0}\text{°};-\text{0.27})$$

$$x$$-intercept: $$(\text{30}\text{°};0)$$

Asymptotes: $$\theta = -\text{150}\text{°}$$

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