Discovering the Characteristics of Sine Functions
Discovering the Characteristics of Sine Functions
Domain and range
For \(f(\theta) = a\sin\theta +q\), the domain is \([0°;360°]\)
The range of \(f(\theta) = a\sin\theta +q\) depends on the values of \(a\) and \(q\).
For \(a>0\):
\begin{align*} -1& \le \sin \theta \le 1 \\ -a& \le a\sin\theta \le a \\ -a+q& \le a\sin\theta +q \le a+q \\ -a+q& \le f(\theta) \le a+q \end{align*}For all values of \(\theta\), \(f(\theta)\) is always between \(-a+q\) and \(a+q\).
Therefore for \(a>0\), the range of \(f(\theta) = a\sin\theta +q\) is \(\{f(\theta):f(\theta)\in [-a+q, a+q]\}\)
Similarly, for \(a<0\), the range of \(f(\theta)=a \sin\theta +q\) is \(\{f(\theta):f(\theta)\in [a+q, -a+q]\}\)
Period
The period of \(y=a\sin\theta +q\) is \(360°\). This means that one sine wave is completed in \(360°\).
Intercepts
The \(y\)-intercept of \(f(\theta)=a\sin\theta +q\) is simply the value of \(f(\theta)\) at \(\theta = 0°\)
\begin{align*} y& = f(0°) \\ & = a\sin 0° + q \\ & = a(0) + q \\ & = q \end{align*}This gives the point \((0;q)\)
Important: when sketching trigonometric graphs, always start with the basic graph and then consider the effects of \(a\) and \(q\).
Example
Question
Sketch the graph of \(f(\theta)=2\sin\theta +3\) for \(\theta \in [0°;360°]\).
Examine the standard form of the equation
From the equation we see that \(a>1\) so the graph is stretched vertically. We also see that \(q>0\) so the graph is shifted vertically upwards by \(\text{3}\) units.
Substitute values for \(\theta\)
|
\(\theta\) |
\(0°\) |
\(30°\) |
\(60°\) |
\(90°\) |
\(120°\) |
\(150°\) |
\(180°\) |
\(210°\) |
\(240°\) |
\(270°\) |
\(300°\) |
\(330°\) |
\(360°\) |
|
\(f(\theta)\) |
\(\text{3}\) |
\(\text{4}\) |
\(\text{4.73}\) |
\(\text{5}\) |
\(\text{4.73}\) |
\(\text{4}\) |
\(\text{3}\) |
\(\text{2}\) |
\(\text{1.27}\) |
\(\text{1}\) |
\(\text{1.27}\) |
\(\text{2}\) |
\(\text{3}\) |
Plot the points and join with a smooth curve
Domain: \([0°;360°]\)
Range: \([1;5]\)
\(x\)-intercepts: none
\(y\)-intercepts: \((0°;3)\)
Maximum turning point: \((90°;5)\)
Minimum turning point: \((270°;1)\)
This lesson is part of:
Functions I