Problem 53 - Precalculus: A Concise Course

Question

Sketch the graph of the function. (Include two full periods.) $$ y=2-\sin \frac{2 \pi x}{3} $$

Step-by-Step Solution

Answer

The amplitude of the function is 2, the period is 3, the horizontal shift is 0, and the vertical shift is 0. The graph goes through two complete cycles from a starting point on the midline, then dips to a valley and back up to the midline, then emerges to a peak, and lowers until it reaches the midline again. This repeats for the second cycle.

Step 1: Determine The Amplitude

The amplitude of a sine function is |A|. In this case A = 2, so the amplitude is \(|-2| = 2\). This is the maximum distance the graph will reach from the midline (or vertical shift).

Step 2: Calculate The Period

The period of a sine function is determined as \( \frac{2\pi}{|B|} \). Given that B=2π, the period will be \( \frac{2\pi}{|2\pi|} = 1 \). Multiplying this with D yields the actual period, which is \(1 * 3 = 3\). Thus, the function completes one full cycle every 3 units.

Step 3: Assess The Horizontal Shift

The horizontal shift or phase shift of a sine function is found as \( -\frac{C}{B} \). In this case, since C=0, the horizontal shift is 0.

Step 4: Identify The Vertical Shift

The vertical shift of a sine function is E. In this case, E = 0, hence the midline of the graph is y=0.

Step 5: Sketch The Graph

Using the values for amplitude, period, horizontal shift, and vertical shift, plot two full periods of the graph on a suitable graph paper. The function starts from the midline, goes down to a minimum, comes back to the midline, then to a maximum, and back to the midline again to complete one period. This pattern repeats for the second period.

Key Concepts

Amplitude

When you encounter trigonometric functions like the sine function, one key concept to grasp is amplitude. The amplitude represents how much the graph of the sine function deviates from its midline. For a function in the form of \[y = A \, \sin(Bx + C) + D,\]the amplitude is given by \(|A|\).

• In simple terms, the amplitude is the vertical distance from the midline to the peak or trough of the wave.
• This "+/-" deviation from the midline implies that as the amplitude increases, the wave appears 'taller'.

In our original function \[y = 2 - \sin \left(\frac{2\pi x}{3}\right),\]the coefficient in front of the sine term is indeed calculated to find the amplitude, but observe that here the sine term's coefficient appears negative. Although the negative sign indicates a reflection, the amplitude itself is measured as a magnitude, which remains positive:

• Amplitude = \(|-1| = 1\), not 2 as the constant number outside the sine function doesn't influence the amplitude.

This means that from the midline, the sine wave's highest and lowest points should be one unit above and below, respectively. Understanding amplitude helps visually contrast the trigonometric graph's height.

Period of a Function

The period of a function is related to how often its shape cycles or repeats over a specific interval. For sine functions expressed as \[y = A \, \sin(Bx + C) + D,\]the period can be calculated using the formula:\[\frac{2\pi}{|B|}.\]

• This period indicates the horizontal distance required for the function to complete one full cycle.
• In essence, it's about how many lengths along the x-axis it takes for the sine wave to start repeating its pattern.

For the function given, \[y = 2 - \sin \left(\frac{2\pi x}{3}\right),\]the B value is influenced by the division: \[\frac{2\pi}{3}.\]So, the calculation of the period becomes: \[\frac{2\pi}{\left|\frac{2\pi}{3}\right|} = 3.\]

• This result suggests the sine function resets its pattern every 3 units along the x-axis.
• Thus, to sketch two full periods, you'd need to extend from 0 to 6 on the x-axis.

This does not just help in plotting the graph; understanding the period gives insight into the frequency—the reciprocal of the period, showcasing how frequently oscillations occur.

Vertical Shift

Vertical shift moves the entire graph of a function up or down on the coordinate plane. In a general sine function expressed as \[y = A \, \sin(Bx + C) + D,\]the vertical shift is represented by the D value:

• If D is positive, the midline of the graph will be lifted upwards by D units.
• Conversely, a negative D moves the graph downward.

In our example, \[y = 2 - \sin \left(\frac{2\pi x}{3}\right),\]the entire sine function is being shifted up by 2 units:

• This means the midline of the sine wave, usually at \(y=0\), is now positioned at \(y=2\).

Thus, while the peaks and troughs of the sine wave remain separated by the amplitude, the entire graph is consistently raised by 2 units above the horizontal axis.
This vertical shift reflects how the sine wave's core location changes, impacting the "height" but not the amplitude range (1). Understanding vertical shift is crucial for accurately sketching transformations of trigonometric functions.