Read amplitude, midline and period from the transformed function.
Notes and key methods
For \(\displaystyle y=A \sin (Bx)+D\) with A>0 and B>0, the amplitude is A, the midline is \(\displaystyle y=D\), and the period is \(\displaystyle \frac{2\pi }{B}\). The range is [D−A,D+A].
Sketch one cycle using the zeros of the unshifted sine, its maximum and its minimum, then repeat periodically. A horizontal scale change affects x-coordinates; a vertical shift affects y-coordinates.
Degrees, radians and a tangent graph
For a sine or cosine graph with inner multiplier B, the period is \(\displaystyle \frac{360^{\circ}}{|B|}\) in degrees and \(\displaystyle \frac{2\pi }{|B|}\) in radians. A tangent graph has period \(\displaystyle \frac{180^{\circ}}{|B|}\) or \(\displaystyle \frac{\pi }{|B|}\). Convert all marked inputs using \(180^\circ=\pi\) radians; calculus formulas use radians.
For \(y=3\tan(2x)+4\) in degrees, the period is \(90^\circ\), the midline is \(\displaystyle y=4\), and poles satisfy \(x=45^\circ+90^\circ n\), n an integer. Unlike sine, tangent has no finite amplitude or maximum.
Check the conditions
Angles are radians.
Both A and B are positive here.
Three worked examples
Example 1 · Trigonometric graph parameters
Consider \(y = 3 \sin{\left(2 x \right)} + 2\), with x measured in radians.
Use the labelled scales alongside the numerical method.
State the amplitude, period and range.
Give the coordinates of the first maximum and first minimum for \(\displaystyle x\ge 0\).
Find the period divided by π.
Find the amplitude.
Show the reasoning
(a)
Use the vertical coefficient, horizontal coefficient and midline separately.
Answer: \(3\); \(\pi\); \(\left[-1, 5\right]\)
(b)
The unscaled sine has its first maximum at \(\displaystyle \frac{\pi }{2}\) and minimum at \(\displaystyle \frac{3\pi }{2}\); divide their inputs by B.
The unscaled sine has its first maximum at \(\displaystyle \frac{\pi }{2}\) and minimum at \(\displaystyle \frac{3\pi }{2}\); divide their inputs by B.
Use half the difference between maximum and minimum.
\(4\)
Answer: \(4\).
Example 3 · Trigonometric graph parameters
Consider \(y = 5 \sin{\left(x \right)} + 4\), with x measured in radians.
Use the labelled scales alongside the numerical method.
State the amplitude, period and range.
Give the coordinates of the first maximum and first minimum for \(\displaystyle x\ge 0\).
Find the period divided by π.
Find the amplitude.
Show the reasoning
(a)
Use the vertical coefficient, horizontal coefficient and midline separately.
Answer: \(5\); \(2 \pi\); \(\left[-1, 9\right]\)
(b)
The unscaled sine has its first maximum at \(\displaystyle \frac{\pi }{2}\) and minimum at \(\displaystyle \frac{3\pi }{2}\); divide their inputs by B.
