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
On the same system of axes, plot the following graphs for \(-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}\):
- \(y_1 = \tan \theta\)
- \(y_2 = \tan (\theta – \text{60}\text{°})\)
- \(y_3 = \tan (\theta – \text{90}\text{°})\)
- \(y_4 = \tan (\theta + \text{60}\text{°})\)
- \(y_5 = \tan (\theta + \text{180}\text{°})\)
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
- Sketch the following functions on the same set of axes for \(-\text{180}\text{°} \leq \theta \leq \text{180}\text{°}\).
- \(y_1 = \tan \theta\)
- \(y_2 = \tan (\theta + \text{30}\text{°})\)
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 \((\text{0}\text{°};\tan p)\).