Discovering the Characteristics of Tangent Functions
Discovering the Characteristics of Tangent Functions
Domain and range
From the graph we see that \(\tan\theta\) is undefined at \(\theta =90°\) and \(\theta =270°\).
Therefore the domain is \(\{\theta :0°\le \theta \le 360°, \theta \ne 90°; 270°\}\).
The range is \(\{f(\theta):f(\theta)\in \mathbb{R}\}\).
Period
The period of \(y=a\tan\theta +q\) is \(180°\). This means that one tangent cycle is completed in \(180°\).
Intercepts
The \(y\)-intercept of \(f(\theta)=a\tan\theta +q\) is simply the value of \(f(\theta)\) at \(\theta =0°\).
\begin{align*} y& = f(0°) \\ & = a\tan 0°+q \\ & = a(0)+q \\ & = q \end{align*}This gives the point \((0°;q)\).
Asymptotes
The graph has asymptotes at \(\theta = 90°\) and \(\theta =270°\).
Example
Question
Sketch the graph of \(y=2\tan\theta +1\) for \(\theta \in [0°;360°]\).
Examine the standard form of the equation
We see that \(a>1\) so the branches of the curve will be steeper. We also see that \(q>0\) so the graph is shifted vertically upwards by \(\text{1}\) unit.
Substitute values for \(\theta\)
|
\(\theta\) |
\(0°\) |
\(30°\) |
\(60°\) |
\(90°\) |
\(120°\) |
\(150°\) |
\(180°\) |
\(210°\) |
\(240°\) |
\(270°\) |
\(300°\) |
\(330°\) |
\(360°\) |
|
\(y\) |
\(\text{1}\) |
\(\text{2.15}\) |
\(\text{4.46}\) |
– |
\(-\text{2.46}\) |
\(-\text{0.15}\) |
\(\text{1}\) |
\(\text{2.15}\) |
\(\text{4.46}\) |
– |
\(-\text{2.46}\) |
\(-\text{0.15}\) |
\(\text{1}\) |
Plot the points and join with a smooth curve
Domain: \(\{\theta :0°\le \theta \le 360°,\theta \ne 90°;270°\}\)
Range: \(\{f(\theta):f(\theta)\in \mathbb{R}\}\)
This lesson is part of:
Functions I