Determining rules for graphs of trig functions
Theory
Determining the rule of a sinusoidal graph means reading four features and writing \(y=a\sin\!\big(n(t-e)\big)+b\) (or a cosine): the amplitude \(a\) from the distance between the midline and a peak, the mean \(b\) as the midline, the period to get \(n=\dfrac{2\pi}{\text{period}}\), and the phase shift \(e\) from a reference point. Choosing sine or cosine depends on which reference point is easiest to read. Radians throughout.
A sinusoid is any curve of the form \(y=a\sin\!\big(n(t-e)\big)+b\) or \(y=a\cos\!\big(n(t-e)\big)+b\). Determining its rule means reading the four constants \(a\), \(b\), \(n\) and \(e\) directly from the graph. Each constant controls one feature: \(a\) the height of the swing, \(b\) the vertical position, \(n\) how fast it repeats, and \(e\) how far it is shifted sideways.
The mean \(b\) is the value halfway between the maximum and minimum — the height of the horizontal midline the curve oscillates about: \(b=\dfrac{\text{max}+\text{min}}{2}\). The amplitude \(a\) is the distance from that midline up to a peak, which is half the full peak-to-trough distance: \(a=\dfrac{\text{max}-\text{min}}{2}\). Amplitude is always positive.
The period is the horizontal length of one complete cycle; from it \(n=\dfrac{2\pi}{\text{period}}\). The phase shift \(e\) is read from a reference point: for a sine rule, \(e\) is where the curve crosses the midline going upward; for a cosine rule, \(e\) is where a maximum occurs. Either function can fit any sinusoid, so pick the one whose reference point is easiest to locate.
The general sinusoid, in factored form so the shift is read directly:
Amplitude and mean from the maximum and minimum values:
The dilation factor \(n\) from the period, and the period from \(n\):
How to determine the rule of a sinusoidal graph
- Mean \(b\). Read the maximum and minimum, then \(b=\dfrac{\text{max}+\text{min}}{2}\). This is the midline the curve swings about.
- Amplitude \(a\). Take \(a=\dfrac{\text{max}-\text{min}}{2}\), the distance from the midline to a peak (always positive).
- Dilation \(n\). Measure the period (one full cycle) and use \(n=\dfrac{2\pi}{\text{period}}\).
- Choose sine or cosine, find \(e\). Pick cosine if a maximum sits at a convenient \(t\) (then \(e\) is that \(t\)); pick sine if an upward midline crossing is clear (then \(e\) is that \(t\)).
- Write and check. Assemble \(y=a\sin\!\big(n(t-e)\big)+b\) and substitute a known point to confirm it fits.
| \(b\) | \(=\) | \(\dfrac{7+1}{2}=4\) |
| \(a\) | \(=\) | \(\dfrac{7-1}{2}=3\) |
| \(n\) | \(=\) | \(\dfrac{2\pi}{2\pi}=1\) |
| \(n\) | \(=\) | \(\dfrac{2\pi}{\text{period}}\) |
| \(=\) | \(\dfrac{2\pi}{\pi/2}=4\) |
| \(y\) | \(=\) | \(2\sin(4t)+0\) |
| \(a=2,\ b\) | \(=\) | \(0\) |
| \(n\) | \(=\) | \(\dfrac{2\pi}{2\pi}=1\) |
| \(y\) | \(=\) | \(2\sin\!\big(1\cdot(t-\dfrac{\pi}{6})\big)+0\) |
| \(b\) | \(=\) | \(\dfrac{4+(-2)}{2}=1\) |
| \(a\) | \(=\) | \(\dfrac{4-(-2)}{2}=3\) |
| \(n\) | \(=\) | \(\dfrac{2\pi}{\pi}=2\) |
| \(y\) | \(=\) | \(3\cos\!\big(2(t-\dfrac{\pi}{3})\big)+1\) |
Common pitfalls
Frequently asked questions
How do you find the amplitude of a sinusoid from its graph?
The amplitude is half the distance between the maximum and minimum: \(a=\dfrac{\text{max}-\text{min}}{2}\). For a maximum of \(5\) and minimum of \(1\), \(a=\dfrac{5-1}{2}=2\). It is always taken positive.
How do you find the mean or midline of a sinusoid?
The mean is halfway between the maximum and minimum: \(b=\dfrac{\text{max}+\text{min}}{2}\). This is the midline \(y=b\) the curve oscillates about; for a max of \(5\) and min of \(1\), \(b=3\).
How do you find n from the period of the graph?
Measure the period (one full cycle) and use \(n=\dfrac{2\pi}{\text{period}}\). A period of \(\pi\) gives \(n=2\); a period of \(\dfrac{\pi}{2}\) gives \(n=4\).
How do you find the phase shift of a sinusoid?
Write it as \(y=a\sin\!\big(n(t-e)\big)+b\). For sine, \(e\) is an upward midline crossing; for cosine, \(e\) is a maximum. It is the shift to the right.
Should I use sine or cosine when determining the rule?
Either fits any sinusoid, so choose the easier reference point. Use cosine when a maximum sits at a convenient \(t\); use sine when a clear upward midline crossing is visible.
How do you check a sinusoidal rule is correct?
Substitute a known point into your rule and confirm the value, and check the amplitude, mean and period match the graph. Testing a maximum or a midline crossing verifies all four constants.