Problem 35 - Precalculus with Calculus Previews

Question

(a) sketch one cycle of the graph of the given function. (b) Find the amplitude \(|A|\) by inspection of the function. (c) Find the maximum value \(M\) and the minimum value \(m\) of the function on the interval in part (a). (d) Then use (4) to verify the amplitude \(|A|\) of the function. (e) Give the range of each function. $$ y=3-2 \sin \pi x $$

Step-by-Step Solution

Answer

Amplitude is 2; Maximum is 5, Minimum is 1; Range is [1, 5].

Step 1: Identify the Function Properties

The given function is \( y = 3 - 2 \sin \pi x \). It is a vertically shifted and stretched sine function. The basic component is \( -2 \sin \pi x \) which has an amplitude and vertical shift.

Step 2: Sketch One Cycle of the Function

The function \( y = 3 - 2 \sin \pi x \) is a transformation of \( \sin x \). The period of \( \sin \pi x \) is \( \frac{2\pi}{\pi} = 2 \). Therefore, one complete cycle occurs from \( x = 0 \) to \( x = 2 \). Sketch it starting from \( y = 3 \), reaching a minimum at \( y = 1 \) when \( \sin \pi x = 1 \), a maximum at \( y = 5 \) when \( \sin \pi x = -1 \), and returning to \( y = 3 \) at \( x = 2 \).

Step 3: Find Amplitude by Inspection

From the function \( y = 3 - 2 \sin \pi x \), the amplitude can be found by taking the absolute value of the coefficient of the sine function which is \( |-2| = 2 \).

Step 4: Determine Maximum and Minimum Values

The maximum value \( M \) occurs when \( \sin \pi x = -1 \), thus \( y = 3 - 2(-1) = 5 \). The minimum value \( m \) occurs when \( \sin \pi x = 1 \), thus \( y = 3 - 2(1) = 1 \).

Step 5: Verify the Amplitude

Using the formula for amplitude, max - min divide by 2: \( \frac{5 - 1}{2} = 2 \), confirming that the amplitude found by inspection is correct.

Step 6: Determine the Range of the Function

The range of the function \( y = 3 - 2 \sin \pi x \) is between its minimum value 1 and maximum value 5, thus, the range is \([1, 5]\).

Key Concepts

Amplitude

In trigonometric functions, the amplitude is the height from the centerline to the peak or trough of the wave. It describes how much the sine wave reaches its highest and lowest points compared to the horizontal midline.
The amplitude is always the absolute value of the coefficient in front of the sine function. For example, in our exercise with the function \[ y = 3 - 2 \sin \pi x \] we can easily determine that the amplitude is \[|A| = |-2| = 2\].

The larger the amplitude, the taller the peaks and deeper the troughs of the wave will be. This information is crucial when sketching graphs and understanding the behavior of trigonometric graphs.

Sine Function

The sine function is one of the fundamental trigonometric functions. It is defined as the y-coordinate of a unit circle's point when an angle is made with the horizontal axis. This function is periodic, meaning it repeats its values in regular intervals.
The standard form of a sine function is \( y = a \sin bx + c\). Each variable affects the graph:

• \( a \) impacts the amplitude.
• \( b \) affects the period.
• \( c \) is a vertical shift.

Our exercise involves the transformation of a sine function defined by \( y = 3 - 2 \sin \pi x \). Here, \( -2 \) reflects a vertical stretch and inversion of the basic \( \sin x \) graph, and \( 3 \) represents a vertical shift upwards. Recognizing these transformations is key to understanding how the graph of the function behaves.

Maximum and Minimum Values

In trigonometric functions like sine, the maximum and minimum values represent the highest and lowest points that the graph reaches within one period. These points are crucial to defining the range of the function.
For the function \( y = 3 - 2 \sin \pi x \),
the maximum occurs when \( \sin \pi x = -1 \), leading to \( y = 5 \). The minimum occurs when \( \sin \pi x = 1 \), resulting in \( y = 1 \).
Determining these values helps us understand how far up or down the graph moves, which is essential for accurately sketching the graph.

Graphing Trigonometric Functions

When graphing trigonometric functions, understanding the transformations such as amplitude, vertical shifts, and period changes is critical. For the provided function \( y = 3 - 2 \sin \pi x \),
we need to account for each of these factors:

• The amplitude \( 2 \) which dictates how high and low the wave goes.
• The vertical shift \( 3 \), moving the baseline of the wave.
• The period given by \( \frac{2\pi}{\pi} = 2 \), indicating how often the wave repeats in length 2.

Graphing involves plotting points over one period, ensuring that the curve peaks at stretch-induced heights, reaches minimum values with proper calculations, and reflects the transformed sine wave's periodic nature. A clear understanding of these components helps make accurate, visually appealing graphs exemplifying the function's complete cycle.