11G - Graphing tangent
Similar to sine and cosine, \(y=tan(x)\) is a many-to-one function with periodic behaviour.
Period and asymptotes:
Consider the rule \(y=A\times tan(nx)\).
Consider the rule \(y=A\times tan(nx)\).
- The period is the length of the interval for one cycle of the function. The period is given by \(\frac{\pi}{n}\).
- The amplitude is meaningless for the tangent function.
- For \(y=A\times tan(n(x-b))+c\), the asymptotes are located at \( n(x-b) = \pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, ...\)
- Asymptotes must be labelled with their equation.
Sketching the tangent function with transformations:
Consider the equation \(y=A\times tan(n(x-b))+c\).
Consider the equation \(y=A\times tan(n(x-b))+c\).
- \(A\) causes a dilation by a factor of \(A\) from the \(x\)-axis. If \(A<0\) then the graph is reflected over the \(x\)-axis. Dilations from the \(x\)-axis are dealt with by considering the amplitude of the graph.
- \(n\) causes a dilation by a factor of \(\frac{1}{n}\) from the \(y\)-axis. If \(n<0\) then the graph is reflected over the \(y\)-axis. Dilations from the \(y\)-axis are dealt with by considering the period of the graph.
- \(b\) causes a horizontal translation of \(b\) units parallel to the \(x\)-axis.
- \(c\) causes a vertical translation of \(c\) units parallel to the \(y\)-axis.
11G - Example 1: Sketching the tangent function
Sketch the graph of \(y=3 tan(x)-\sqrt{3}\) for \(x \in [-2\pi,2\pi]\). Clearly label all asymptotes with their equations and axial intercepts with their coordinates.
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11G - Example 1: Video solution
11G - Example 1: Practice
Question 1: ABC Question 2: ABC 11G - Example 1: Solutions
Question 1: ABC Question 2: ABC |
11G - Example 2: Sketching the tangent function
Sketch the graph of \(y=-\frac{1}{2}tan(2x-\frac{\pi}{3})\) for \(x \in [-\pi,\pi]\). Clearly label all asymptotes with their equations and axial intercepts with their coordinates.
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11G - Example 2: Video solution
11G - Example 2: Practice
Question 1: ABC Question 2: ABC 11G - Example 2: Solutions
Question 1: ABC Question 2: ABC |
Determining the rule of a tangent graph:
11G - Example 3: Determining the rule of a tangent graph
State the equation of the image graph is \(y=tan(x)\) undergoes the following transformations:
■ Please note that transformations of graphs were covered in detail in Section 5. |
11G - Example 3: Video solution
11G - Example 3: Practice
Question 1: ABC Question 2: ABC 11G - Example 3: Solutions
Question 1: ABC Question 2: ABC |
11G - Example 4: Determining the rule of a tangent graph
The graph shown below can be expressed as \(y=tan(nx)+c\), where \(n,c \in \mathbb{R}\).
state a suitable set of values for \(n\) and \(c\).
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11G - Example 4: Video solution
11G - Example 4: Practice
Question 1: Sketch the graph of \(y=3tan(4x)+3\sqrt{3}\) for \(x\in [0,\frac{\pi}{2}]\). Clearly label any axial intercepts and endpoints with their coordinates and asymptotes with their equations. Question 2: ABC 11G - Example 4: Solutions
Question 1: ABC Question 2: ABC |