Question 6.4.7: Graphing y = a sin [b(x - c)] and y = a cos [b(x - c)] OBJEC...
Graphing y = a sin [b(x – c)] and y = a cos [b(x – c)]
OBJECTIVE
Graph a function of the form y = a sin [b(x – c)] or y = a cos [b(x – c)], with b > 0, by finding the amplitude, period, and phase shift.
Step 1 Find the amplitude, period, and phase shift.
amplitude = |a|
period = \frac{2π}{b}
phase shift = c
If c > 0, shift to the right.
If c < 0, shift to the left.
Step 2 The cycle begins at x = c. One complete cycle occurs over the interval [c,c+\frac{2π}{b}].
Step 3 Divide the interval [c,c+\frac{2π}{b}] into four equal parts, each of length \frac{1}{4}(\text { period }) = \frac{1}{4}(\frac{2π}{b}) This gives the x-coordinates for the five key points:
c, c + \frac{1}{4}(\frac{2π}{b}), c + \frac{1}{2}(\frac{2π}{b}) ,\\ c + \frac{3}{4}(\frac{2π}{b}), \text{and }c + \frac{2π}{b}Step 4 If z > 0, for y = a sin [b(x – c)], sketch one cycle of the sine curve through the key points (c, 0),
(c + \frac{π}{2b},a), (c + \frac{π}{b},0), \\(c + \frac{3π}{2b},-a), \text{and } (c + \frac{2π}{b},0),For y = a cos [b(x – c)], sketch one cycle of the cosine curve through the key
points (c, a), (c + \frac{π}{2b},0), (c + \frac{π}{b},-a), \\(c + \frac{3π}{2b},0), \text{and } (c + \frac{2π}{b},a),
If a < 0, reflect the graph of
y = |a| sin b(x – c) or
y = |a| cos b(x – c) in the x-axis.
Find the amplitude, period, and phase shift and sketch the graph of y=3 \sin \left[2\left(x-\frac{\pi}{4}\right)\right] over a one-period interval.
1.
\begin {array}{rr} y = & 3 & \sin \left[2 \left(x – \frac{\pi}{4} \right) \right] \\ & \uparrow & \uparrow \uparrow \\ & a = 3, & b = 2, c = \frac{\pi}{4} \end {array}
amplitude = |3| = 3
period = \frac{2 \pi}{2}=\pi
phase shift = \frac{\pi}{4} \quad \text { Shift right because } \frac{\pi}{4}>0 \text {. }
2. Begin the cycle at x=\frac{\pi}{4} and graph one cycle over \left[\frac{\pi}{4}, \frac{\pi}{4}+\pi\right]=\left[\frac{\pi}{4}, \frac{5 \pi}{4}\right]
3. Divide the interval \left[\frac{\pi}{4}, \frac{5 \pi}{4}\right] into four equal parts, each having length \frac{1}{4}(\text { period })=\frac{1}{4}(\pi)=\frac{\pi}{4} \text {. }
The x-coordinates of the five key points are
\frac{\pi}{4}, \frac{\pi}{4}+\frac{\pi}{4}=\frac{\pi}{2}, \frac{\pi}{4}+\frac{\pi}{2}=\frac{3 \pi}{4}, \frac{\pi}{4}+\frac{3 \pi}{4}=\pi, and
\frac{\pi}{4}+\pi=\frac{5 \pi}{4} .
4. Sketch one cycle of the sine curve through the five key points: \left(\frac{\pi}{4}, 0\right),\left(\frac{\pi}{2}, 3\right),\left(\frac{3 \pi}{4}, 0\right),(\pi,-3), \text { and }\left(\frac{5 \pi}{4}, 0\right).
