Mathematics » Functions II » The Tangent Function

# Functions of the Form y = tan(θ + p)

## Functions of the form $$y=\tan(\theta +p)$$

We now consider tangent functions of the form $$y = \tan(\theta + p)$$ and the effects of parameter $$p$$.

## Optional Investigation: The effects of $$p$$ on a tangent graph

1. On the same system of axes, plot the following graphs for $$-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}$$:

1. $$y_1 = \tan \theta$$
2. $$y_2 = \tan (\theta – \text{60}\text{°})$$
3. $$y_3 = \tan (\theta – \text{90}\text{°})$$
4. $$y_4 = \tan (\theta + \text{60}\text{°})$$
5. $$y_5 = \tan (\theta + \text{180}\text{°})$$
2. Use your sketches of the functions above to complete the following table:

 $$y_1$$ $$y_2$$ $$y_3$$ $$y_4$$ $$y_5$$ period domain range $$y$$-intercept(s) $$x$$-intercept(s) asymptotes effect of $$p$$

### The effect of the parameter on $$y = \tan(\theta + p)$$

The effect of $$p$$ on the tangent function is a horizontal shift (or phase shift); the entire graph slides to the left or to the right.

• For $$p > 0$$, the graph of the tangent function shifts to the left by $$p$$.

• For $$p < 0$$, the graph of the tangent function shifts to the right by $$p$$.

 $$p > 0$$ $$p < 0$$    ## Example

### Question

1. Sketch the following functions on the same set of axes for $$-\text{180}\text{°} \leq \theta \leq \text{180}\text{°}$$.
1. $$y_1 = \tan \theta$$
2. $$y_2 = \tan (\theta + \text{30}\text{°})$$
2. For each function determine the following:

• Period
• Domain and range
• $$x$$- and $$y$$-intercepts
• Asymptotes

### Examine the equations of the form $$y = \tan (\theta + p)$$

Notice that for $$y_1 = \tan \theta$$ we have $$p = \text{0}\text{°}$$ (no phase shift) and for $$y_2 = \tan (\theta + \text{30}\text{°})$$ we have $$p = \text{30}\text{°} > 0$$ and therefore the graph shifts to the left by $$\text{30}$$$$\text{°}$$.

### Complete a table of values

 θ $$-\text{180}$$$$\text{°}$$ $$-\text{135}$$$$\text{°}$$ $$-\text{90}$$$$\text{°}$$ $$-\text{45}$$$$\text{°}$$ $$\text{0}$$$$\text{°}$$ $$\text{45}$$$$\text{°}$$ $$\text{90}$$$$\text{°}$$ $$\text{135}$$$$\text{°}$$ $$\text{180}$$$$\text{°}$$ $$\tan \theta$$ $$\text{0}$$ $$\text{1}$$ undef $$-\text{1}$$ $$\text{0}$$ $$\text{1}$$ undef $$-\text{1}$$ $$\text{0}$$ $$\tan (\theta + \text{30}\text{°})$$ $$\text{0.58}$$ $$\text{3.73}$$ $$-\text{1.73}$$ $$-\text{0.27}$$ $$\text{0.58}$$ $$\text{3.73}$$ $$-\text{1.73}$$ $$-\text{0.27}$$ $$\text{0.58}$$

### Sketch the tangent graphs ### Complete the table

 $$y_1 = \tan \theta$$ $$y_2 = \tan (\theta + \text{30}\text{°})$$ period $$\text{180}\text{°}$$ $$\text{180}\text{°}$$ domain $$\{ \theta: -\text{180}\text{°} \leq \theta \leq \text{180}\text{°}, \theta \ne -\text{90}\text{°}; \text{90}\text{°} \}$$ $$\{ \theta: -\text{180}\text{°} \leq \theta \leq \text{180}\text{°}, \theta \ne -\text{120}\text{°}; \text{60}\text{°} \}$$ range $$(-\infty;\infty)$$ $$(-\infty;\infty)$$ $$y$$-intercept(s) $$(\text{0}\text{°};0)$$ $$(\text{0}\text{°};\text{0.58})$$ $$x$$-intercept(s) $$(-\text{180}\text{°};0)$$, $$(\text{0}\text{°};0)$$ and $$(\text{180}\text{°};0)$$ $$(-\text{30}\text{°};0) \text{ and } (\text{150}\text{°};0)$$ asymptotes $$\theta = -\text{90}\text{°} \text{ and } \theta = \text{90}\text{°}$$ $$\theta = -\text{120}\text{°} \text{ and } \theta = \text{60}\text{°}$$

### Discovering the characteristics

For functions of the general form: $$f(\theta) = y =\tan (\theta + p)$$:

Domain and range

The domain of one branch is $$\{ \theta: \theta \in (-\text{90}\text{°} – p; \text{90}\text{°} – p) \}$$ because the function is undefined for $$\theta = -\text{90}\text{°} – p$$ and $$\theta = \text{90}\text{°} – p$$.

The range is $$\{ f(\theta): f(\theta) \in \mathbb{R} \}$$.

Intercepts

The $$x$$-intercepts are determined by letting $$f(\theta) = 0$$ and solving for $$\theta$$.

The $$y$$-intercept is calculated by letting $$\theta = \text{0}\text{°}$$ and solving for $$f(\theta)$$. \begin{align*} y &= \tan (\theta + p) \\ &= \tan (\text{0}\text{°} + p) \\ &= \tan p \end{align*} This gives the point $$(\text{0}\text{°};\tan p)$$.