Problem 65 - Precalculus with Calculus Previews

Question

Describe in words how you would obtain the graph of the given function by starting with the graph of \(y=\sin x\) (Problem 65 ) and the graph of \(y=\cos x(\) Problem 66\()\). $$ y=5+3 \sin (2 x-\pi) $$

Step-by-Step Solution

Answer

Stretch vertically by 3, compress horizontally by 1/2, shift right \(\frac{\pi}{2}\), and move up 5 units.

Step 1: Identify the Base Function

The base function here is \(y = \sin x\). The given function has modifications including amplitude, horizontal stretch/compression, phase shift, and vertical shift.

Step 2: Apply the Amplitude Change

The amplitude of \(y = \sin x\) is changed from 1 to 3. This means the new function, \(y = 3\sin x\), will stretch vertically by a factor of 3, making its maximum value 3 and minimum value -3.

Step 3: Apply the Horizontal Stretch/Compression

The given function has \(2x\) instead of \(x\). This means the function is horizontally compressed by a factor of \(\frac{1}{2}\). The period of \(y = \sin x\) is \(2\pi\), so the period of \(y = \sin(2x)\) is \(\frac{2\pi}{2} = \pi\).

Step 4: Apply the Phase Shift

The expression inside the sine function is \((2x - \pi)\), indicating a horizontal phase shift. Solve \(2x - \pi = 0\) to find the shift: \(x = \frac{\pi}{2}\). This means the graph shifts \(\frac{\pi}{2}\) units to the right.

Step 5: Apply the Vertical Shift

Finally, the entire function \(3\sin(2x - \pi)\) is shifted upward by 5 units, resulting in \(y = 5 + 3\sin(2x - \pi)\). This shifts the midline from 0 to 5.

Key Concepts

Amplitude Change

When dealing with trigonometric functions like sine and cosine, the amplitude reflects the height of the wave from its centerline. In the base function, \( y = \sin x \), the amplitude is 1, indicating the wave reaches a maximum of 1 and a minimum of -1. However, for the function \( y = 3\sin x \), the amplitude changes to 3.
This transformation stretches the wave vertically:

• The peaks of the wave now reach up to 3.
• The valleys dip down to -3.

This means that every point on the \( \sin x \) curve is multiplied by 3, making the entire wave taller, while keeping the same frequency of oscillations. By controlling the amplitude, we can adjust how pronounced the peaks and troughs of the wave are, impacting the graph significantly.

Horizontal Compression

The term \( 2x \) in the expression \( y = 3\sin(2x - \pi) \) causes a horizontal compression of the graph. Normally, the graph of \( y = \sin x \) has a period of \( 2\pi \), which is the distance over which the wave pattern repeats.
In \( y = \sin(2x) \), the factor 2 inside the argument compresses the graph:

• The new period is \( \frac{2\pi}{2} = \pi \).
• This means the wave repeats twice as often within the same horizontal space.

Understanding horizontal compression is about seeing how frequently the wave peaks and troughs in a given length. This compressing effect essentially "squeezes" the wave, reducing the distance between corresponding points on successive cycles.

Phase Shift

A phase shift moves the entire wave along the horizontal axis. In \( y = 3\sin(2x - \pi) \), the phase shift is determined by the \( -\pi \) term inside the function.
Solve \( 2x - \pi = 0 \) to find the shift:

• Resulting in \( x = \frac{\pi}{2} \).
• This indicates a shift of \( \frac{\pi}{2} \) units to the right.

Phase shifts are helpful for aligning waves with different phases or adjusting when certain features, like peaks or crossings, appear along the x-axis. By moving the graph this way, you can synchronize it with other cyclical phenomena or adjust peak timings.

Vertical Shift

The vertical shift affects the entire graph by moving it up or down along the y-axis. In the function \( y = 5 + 3\sin(2x - \pi) \), the whole wave is shifted upwards by 5 units.
This modification changes the midline (average height) from 0 to 5:

• All points on the wave are increased by 5 units.
• This raises the means for the maximum from 3 to 8, and the minimum from -3 to 2.

Vertical shifts are useful for setting a new baseline for oscillations, ensuring that the graph aligns with or reaches desired levels. This fundamental change in position plays a critical role when comparing multiple waves or ensuring a starting offset.