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| (a) |
The maximum value is \(1\) while the minimum value is \(-1\),
so the amplitude of the graph is \(1\) unit.
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| (b) |
The graph repeats itself every \(360^{\circ}\text{ or }2\pi \text{ rad}\),
so \(360^{\circ}\text{ or }2\pi \text{ rad}\) is the period for both graphs.
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| (a) |
This graph has no maximum or minimum value.
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| (b) |
The graph repeats itself every \(180^{\circ}\text{ or }\pi \text{ rad}\) interval,
so the period of a tangent graph is \(180^{\circ}\text{ or }\pi \text{ rad}\).
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| (c) |
The function \(y = \text{tan } x\) is not defined at \(x = 90^{\circ} \text{ and } x = 270^{\circ}\).
The curve approaches the line but does nottouch the line.
This line is called an asymptote.
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- The graphs for these three functions are as follows:
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Graph \(y = \text{cos } x\) for \(-2\pi \leqslant x \leqslant 2\pi\) |
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| (a) |
Amplitude \(=1\)
- The maximum value of \(y=1\)
- The minimum value of \(y=-1\)
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| (b) |
Period \(=360^{\circ}\text{ or }2\pi \) |
| (c) |
\(x\)-intercepts:
\(-\dfrac{3}{2}\pi, \ -\dfrac{1}{2}\pi, \ \dfrac{1}{2}\pi, \ \dfrac{3}{2}\pi\)
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| (d) |
\(y\)-intercepts: \(1\) |
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Graph \(y = \text{sin } x\) for \(-2\pi \leqslant x \leqslant 2\pi\)
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| (a) |
Amplitude \(=1\)
- The maximum value of \(y=1\)
- The minimum value of \(y=-1\)
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| (b) |
Period \(=360^{\circ}\text{ atau }2\pi \) |
| (c) |
\(x\)-intercepts:
\(-2\pi, \ -\pi, \ 0, \ \pi, \ 2\pi \)
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| (d) |
\(y\)-intercepts: \(0\) |
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Graph \(y = \text{tan } x\) for \(-2\pi \leqslant x \leqslant 2\pi\) |
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| (a) |
No amplitude
- There are no maximum and minimum values of \(y\)
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| (b) |
Period \(=180^{\circ}\text{ or }\pi \) |
| (c) |
\(x\)-asymptotes:
\(-\dfrac{3}{2}\pi, \ -\dfrac{1}{2}\pi, \ \dfrac{1}{2}\pi, \ \dfrac{3}{2}\pi\)
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| (d) |
\(x\)-intercepts:
\(-2\pi, \ -\pi, \ 0, \ \pi, \ 2\pi \)
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| (e) |
\(y\)-intercepts: \(0\) |
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The values of \(a\), \(b\) and \(c\) in the function \(y = a \text{ sin } bx + c\) affect the amplitude, the period and the position of the graph
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The effects of changing the values of \(a\), \(b\) and \(c\) on the graph can be summarised as follows:
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| Change in |
Effects |
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The maximum and minimum values of the graphs (except for the graph of \(y = \text{tan } x\) where there is no maximum or minimum value)
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Number of cycles in the range \(0^{\circ} \leqslant x \leqslant 360^{\circ} \text{ or }0^{\circ} \leqslant x \leqslant 2\pi\):
- Graphs \(y = \text{sin } x\) and \(y = \text{cos } x\) \(\begin{pmatrix} \text{period } = \dfrac{360^{\circ}}{b} \text{ or } \dfrac{2}{b}\pi\end{pmatrix}\)
- Graph \(y = \text{tan } x \ \begin{pmatrix} \text{period } = \dfrac{180^{\circ}}{b} \text{ or } \dfrac{1}{b}\pi\end{pmatrix}\)
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The position of the graph with reference to the \(x\)-axis as compared to the position of the basic graph |
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Example:
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| Example |
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State the cosine function represented by the graph above.
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Solution: |
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Note that the amplitude is \(4\).
So, \(a=4.\)
Two cycles in the range of \(0^{\circ} \leqslant x \leqslant 2\pi\).
The period is \(\pi\), that is, \(\dfrac{2\pi}{b} = \pi, \text{ so }b=2.\)
Hence, the graph represents \(y= 4\text{ cos }2x\).
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