Sketching of \(y=a\sin \left( bx \right)+c\) and \(y=a\cos \left( bx \right)+c\)
Sketching of \(y=a\sin \left( bx \right)+c\) and \(y=a\cos \left( bx \right)+c\)
For \(y=a\sin(bx)+c\) or \(y=a\cos(bx)+c\), \(a,b\ne0\):
\(\text{Midline:}\hspace{0.5em}y=c\qquad\text{Amplitude}=|a|\qquad T=\dfrac{2\pi}{|b|}\text{ or}\hspace{0.5em}\dfrac{360^\circ}{|b|}\)
\(y_{\max}=c+|a|\qquad y_{\min}=c-|a|\)
The sketches below use \(a,b>0\).
| Equation | Midline | Amplitude | Maximum and minimum | Period |
|---|---|---|---|---|
| \(y=3\sin 2x+1\) | ||||
| \(y=5-2\cos\frac{x}{2}\) |
Example: \(y=2\cos3x-1\), \(0\le x\le\pi\)
| Feature | Working |
|---|---|
| Midline | |
| Amplitude | |
| Maximum and minimum | |
| Period | |
| Start point |
Mark the midline and extremes, plot at quarter-period intervals, then stop at both endpoints.
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