The constants in \(y=a\sin bx+c\) change the base graph in three predictable ways: \(a\) stretches it vertically, \(b\) squeezes it horizontally, and \(c\) slides it up or down. Read all three straight from the equation.
Features of \(y=a\sin bx+c\), \(y=a\cos bx+c\) and \(y=a\tan bx+c\) for \(a,b>0\)
| Feature | \(y=a\sin bx+c\) or \(y=a\cos bx+c\) | \(y=a\tan bx+c\) |
|---|---|---|
| Midline | \(y=c\) | \(y=c\) |
| Amplitude | \(a\) | No amplitude |
| Period | \(\dfrac{360^\circ}{b}\) or \(\dfrac{2\pi}{b}\) | \(\dfrac{180^\circ}{b}\) or \(\dfrac{\pi}{b}\) |
| Maximum and minimum | \(c+a\) and \(c-a\) | No maximum or minimum |
\(y=2\sin x\): the value of \(a\) doubles the height — amplitude \(2\), same period.
\(y=\sin 2x\): the value of \(b\) doubles the repeats — period \(\dfrac{2\pi}{2}=\pi\), same amplitude.
\(y=\sin x+1\): the value of \(c\) lifts the curve — midline \(y=1\), same amplitude and period.
Reading the features from an equation
| Equation | Midline | Amplitude | Max and min | Period |
|---|---|---|---|---|
| \(y=3\sin 2x+1\) | \(y=1\) | \(3\) | \(1+3=4\) \(1-3=-2\) | \(\dfrac{360^\circ}{2}=180^\circ\) \(\dfrac{2\pi}{2}=\pi\) |
| \(y=5-2\cos\dfrac{x}{2}\) | \(y=5\) | \(2\) | \(5+2=7\) \(5-2=3\) | \(\dfrac{360^\circ}{\left(\frac{1}{2}\right)}=720^\circ\) \(\dfrac{2\pi}{\left(\frac{1}{2}\right)}=4\pi\) |
| \(y=3\tan 2x-1\) | \(y=-1\) | No amplitude | None | \(\dfrac{180^\circ}{2}=90^\circ\) \(\dfrac{\pi}{2}\) |
A negative value of \(a\) reflects the curve in its midline \(y=c\): the graph of \(y=-a\sin bx+c\) is the mirror image of \(y=a\sin bx+c\), with the same midline, amplitude and period.
Amplitude is a positive distance. For \(y=2-3\cos 2x\) the amplitude is \(3\), not \(-3\) — the minus sign only reflects the curve in the midline \(y=2\).
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