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 \(p\).

\(p > 0\)\(p < 0\)
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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

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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 \((\text{0}\text{°};\tan p)\).

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