Graphs of Trigonometric Functions
When graphing trigonometric functions, follow these systematic steps:
- Determine the fundamental period of the function.
- Select an appropriate interval matching the calculated period.
- Construct a sign and behavior chart (table of values) for the function over the chosen interval.
- Plot the function’s graph within the selected interval. Replicate the curve identically across subsequent intervals of the fundamental period length.
1. Graph of the Cosine Function:
The fundamental period of the function \( f(x) = \cos x \) is \( 2\pi \). Let us sketch the graph by constructing a table of values on the interval \( [0, 2\pi] \).
\[
\begin{array}{c |lcr}
x & 0 & \displaystyle \ \;\; \frac{\pi}{2} & \pi & \displaystyle \; \; \frac{3\pi}{2} & 2\pi \\
\hline
\cos x & 1 & \searrow \ 0 & \searrow \quad -1 & \nearrow \quad 0 & \nearrow \quad 1
\end{array}
\]
2. Graph of the Sine Function:
The fundamental period of the function \( f(x) = \sin x \) is \( 2\pi \). Let us sketch the graph by constructing a table of values on the interval \( [0, 2\pi] \).
\[
\begin{array}{c|lcr}
x & 0 & \displaystyle \;\;\;\; \frac{\pi}{2} & \pi & \displaystyle \;\; \frac{3\pi}{2} & 2\pi \\
\hline
\sin x & 0 & \nearrow \quad 1 & \searrow \quad 0 & \searrow \quad -1 & \nearrow \quad 0
\end{array}
\]
3. Graph of the Tangent Function:
The fundamental period of the function \( f(x) = \tan x \) is \( \pi \).
Since \( f(x) = \tan x \) is undefined at \( \displaystyle x = \frac{\pi}{2} \), the graph does not intersect the vertical asymptote \( \displaystyle x = \frac{\pi}{2} \).
Let us construct a table of values over the interval \( [0, \pi] – \{ \displaystyle \frac{\pi}{2} \} \) to sketch the graph.
\[
\begin{array}{c|ccc}
x & 0 & \; \; \displaystyle \frac{\pi}{4} & \displaystyle \frac{\pi}{2} & \displaystyle \frac{3\pi}{4} & \; \; \pi \\
\hline
\tan x & 0 & \nearrow 1 & \nearrow +\infty || -\infty \nearrow & -1 & \nearrow 0
\end{array}
\]
4. Graph of the Cotangent Function:
The fundamental period of the function \( f(x) = \cot x \) is \( \pi \).
Since \( f(x) = \cot x \) is undefined at \( x = 0 \) and \( x = \pi \), the graph does not intersect the vertical asymptotes \( x = 0 \) and \( x = \pi \).
Let us construct a table of values over the interval \( [0, \pi] – \{ 0, \pi \} \) to sketch the graph.
\[
\begin{array}{c|lll}
x & \quad & 0 & \;\; \displaystyle \frac{\pi}{4} & \;\; \displaystyle \frac{\pi}{2} & \;\; \displaystyle \frac{3\pi}{4} & \;\; & \quad \pi \\
\hline
\cot x & \quad & || +\infty & \searrow 1 & \searrow 0 & \searrow -1 & \searrow -\infty& \quad ||
\end{array}
\]
5. Graph of the Secant Function:
The fundamental period of the function \( \displaystyle f(x) = \sec x = \frac{1}{\cos x} \) is \( 2\pi \).
Since \( f(x) = \sec x \) is undefined at \( \displaystyle x = \frac{\pi}{2} \) and \( \displaystyle x = \frac{3\pi}{2} \), the graph does not intersect the vertical asymptotes \( \displaystyle x = \frac{\pi}{2} \) and \( \displaystyle x = \frac{3\pi}{2} \).
Let us construct a table of values over the interval \( \displaystyle [0, 2\pi] – \{ \frac{\pi}{2}, \frac{3\pi}{2} \} \) to sketch the graph.
\[
\begin{array}{c|lllll}
x & 0 & \displaystyle \frac{\pi}{2} & \pi & \quad \quad \;\; \displaystyle \frac{3\pi}{2} & 2\pi \\
\hline
\sec x & 1 & \nearrow +\infty || -\infty \nearrow & -1 & \searrow -\infty || +\infty \searrow & 1
\end{array}
\]
6. Graph of the Cosecant Function:
The fundamental period of the function \( \displaystyle f(x) = \csc x = \frac{1}{\sin x} \) is \( 2\pi \). Since \( f(x) = \csc x \) is undefined at \( x = 0 \), \( x = \pi \), and \( x = 2\pi \), the graph does not intersect the vertical asymptotes \( x = 0 \), \( x = \pi \), and \( x = 2\pi \).
Let us construct a table of values over the interval \( [0, 2\pi] – \{ 0, \pi, 2\pi \} \) to sketch the graph.
\[
\begin{array}{c|lllll}
x & 0 & \;\;\;\; \displaystyle \frac{\pi}{2} & \quad \quad \quad \pi & \displaystyle \frac{3\pi}{2} & \quad \quad \quad 2\pi \\
\hline
\csc x & || +\infty & \searrow 1 & \nearrow +\infty || -\infty \nearrow & -1 & \searrow -\infty \; \; ||
\end{array}
\]
Example:
Sketch the graph of the function \( f(x) = 1 + \sin x \).
The period of the function \( f(x) = 1 + \sin x \) is \( 2\pi \). Let us construct a table of values over the interval \( [0, 2\pi] \) to sketch the graph.
\[
\begin{array}{c|lcr}
x & 0 & \displaystyle \frac{\pi}{2} & \pi & \displaystyle \frac{3\pi}{2} & 2\pi \\
\hline
1 + \sin x & 1 & \nearrow \quad 2 & \searrow \quad 1 & \searrow \quad 0 & \nearrow \quad 1
\end{array}
\]
Example:
Sketch the graph of the function \( f(x) = -2 \cos x \).
The fundamental period of the function \( f(x) = -2 \cos x \) is \( 2\pi \). Let us sketch its graph by constructing a table of values over the interval \( [0, 2\pi] \).
\[
\begin{array}{c|lcr}
x & 0 & \quad \displaystyle \frac{\pi}{2} & \pi & \displaystyle \frac{3\pi}{2} & 2\pi \\
\hline
-2 \cos x & -2 & \nearrow \quad 0 & \nearrow \quad 2 & \searrow \quad 0 & \searrow \quad -2
\end{array}
\]
The graph of the function \( y = -\cos x \) is a reflection of the graph of \( y = \cos x \) across the x-axis.
Example:
Sketch the graph of the function \( \displaystyle f(x) = \tan \frac{x}{2} \).
The fundamental period of the function \( \displaystyle f(x) = \tan \frac{x}{2} \) is \( 2\pi \).
Since \( \displaystyle f(x) = \tan \frac{x}{2} \) is undefined at \( x = \pi \), the graph does not intersect the vertical asymptote \( x = \pi \). Let us construct a table of values over the interval \( [0, 2\pi] – \{ \pi \} \) to sketch the graph.
\[
\begin{array}{c|lll}
x & 0 & \quad \displaystyle \frac{\pi}{2} & \quad \quad \quad \pi & \displaystyle \frac{3\pi}{2} & \; \; 2\pi \\
\hline
\tan \frac{x}{2} & 0 & \nearrow 1 & \nearrow +\infty || -\infty \nearrow & -1 & \nearrow 0
\end{array}
\]
The graph of the function \( \displaystyle y = \tan \frac{x}{2} \) represents a horizontal stretch of the curve \( y = \tan x \) mapped over a period of \( 2\pi \).
Example:
Sketch the graph of the function \( \displaystyle f(x) = \sin (x \ – \ \frac{\pi}{4}) \).
The fundamental period of the function \( \displaystyle f(x) = \sin (x \ – \ \frac{\pi}{4}) \) is \( 2\pi \).
Let us sketch its graph by constructing a table of values over the interval \( \displaystyle [\frac{\pi}{4}, \frac{9\pi}{4}] \).
\[
\begin{array}{c|lcr}
x & \displaystyle \frac{\pi}{4} & \quad \displaystyle \frac{3\pi}{4} & \; \displaystyle \frac{5\pi}{4} & \; \displaystyle \frac{7\pi}{4} & \; \displaystyle \frac{9\pi}{4} \\
\hline
\sin (x – \frac{\pi}{4}) & 0 & \nearrow \quad 1 & \searrow \quad 0 & \searrow \quad -1 & \nearrow \quad 0
\end{array}
\]
Example:
Sketch the graph of the function \( f(x) = -2 + \cot(-x) \).
Using the odd function identity, we rewrite the function as \( f(x) = -2 + \cot(-x) = -2 – \cot x \). Its fundamental period is \( \pi \).
Since \( f(x) = -2 – \cot x \) is undefined at \( x = 0 \) and \( x = \pi \), the graph does not intersect the vertical asymptotes \( x = 0 \) and \( x = \pi \).
Let us construct a table of values over the interval \( [0, \pi] – \{ 0, \pi \} \) to sketch the graph.
\[
\begin{array}{c|lcr}
x & 0 & \displaystyle \frac{\pi}{4} & \displaystyle \frac{\pi}{2} & \displaystyle \frac{3\pi}{4} & \pi \\
\hline
-2 – \cot x & || -\infty & \nearrow \quad -3 & \nearrow \quad -2 & \nearrow \quad -1 & \nearrow \quad +\infty ||
\end{array}
\]
The graph of the function \( y = -\cot x \) is a reflection of the graph of \( y = \cot x \) across the x-axis.
The graph of the function \( y = -2 – \cot x \) is a vertical shift of the graph of \( y = -\cot x \) downward by 2 units along the y-axis.
Example:
Sketch the graph of the function \( \displaystyle f(x) = \sin \left( \frac{2x}{3} + \frac{\pi}{3} \right) \).
The fundamental period of the function \( \displaystyle f(x) = \sin \left( \frac{2x}{3} + \frac{\pi}{3} \right) \) is \( 3\pi \).
Let us sketch the graph by constructing a table of values over the interval \( \displaystyle [-\frac{\pi}{2}, \frac{5\pi}{2}] \).
\[
\begin{array}{c|lcr}
x & \displaystyle -\frac{\pi}{2} & \displaystyle \frac{\pi}{4} & \pi & \displaystyle \frac{7\pi}{4} & \displaystyle \frac{5\pi}{2} \\
\hline
\sin (\frac{2x}{3} + \frac{\pi}{3}) & 0 & \nearrow \quad 1 & \searrow \quad 0 & \searrow \quad -1 & \nearrow \quad 0
\end{array}
\]
The graph of the function \( \displaystyle y = \sin \left( \frac{2x}{3} + \frac{\pi}{3} \right) \) can be obtained by shifting the graph of \( \displaystyle y = \sin \frac{2x}{3} \) horizontally to the left along the x-axis by \( \displaystyle \frac{\pi}{2} \) units.