Problem 31 - 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. $$ y=3-4 \cos x $$

Step-by-Step Solution

Answer

One cycle from \(x = 0\) to \(x = 2\pi\), with amplitude 4, maximum 7, minimum -1, and range \([-1, 7]\).

Step 1: Understand the Function

We are given the function \( y = 3 - 4 \cos x \). This is a cosine function with a vertical transformation and vertical shift. Our goal is to complete a set of tasks, including sketching, finding the amplitude, and identifying specific values.

Step 2: Sketch One Cycle of the Graph

To sketch one cycle of the graph of \( y = 3 - 4 \cos x \), begin by identifying the key features of cosine. The function \( y = -4 \cos x \) has a period of \(2\pi\). With the vertical shift of 3, the graph will oscillate around 3. At \( x = 0 \), \( y = -4 \cos 0 + 3 = -4 + 3 = -1 \). At \( x = \pi, \ y = 7 \), and at \( x = 2\pi, \ y = -1 \) again. The critical points are thus marked at intervals of \( \pi \). Draw this oscillatory motion centered at \( y = 3 \) between \( -1 \) and \( 7 \).

Step 3: Find the Amplitude by Inspection

The amplitude of a cosine function \( y = a \cos x + b \) is given by \(|a|\). Here, \( a = -4 \); hence, \(|A| = 4\). The amplitude is the maximum vertical deviation from the mean position.

Step 4: Identify Maximum and Minimum Values

For the function \( y = 3 - 4 \cos x \), the cosine term \( \cos x \) oscillates between -1 and 1. Thus, the minimum value occurs when \( \cos x = 1 \), which makes \( y = -4 + 3 = -1 \). The maximum value is obtained when \( \cos x = -1 \), leading to \( y = 4 + 3 = 7 \).

Step 5: Verify Amplitude with Formula (4)

Formula (4) mentions that the amplitude \(|A|\) equals \(\frac{M-m}{2}\). Using the values from Step 4, with \( M = 7 \) and \( m = -1 \), the amplitude is calculated as \((7 - (-1))/2 = 8/2 = 4\). This confirms our previous finding from Step 3.

Step 6: Determine the Range of the Function

The range of a function is the set of possible output values. Based on the maximum and minimum values calculated, the range of \( y = 3 - 4 \cos x \) is from \(-1\) to \(7\). This can be formally written as \([-1, 7]\).

Key Concepts

Cosine Function

Understanding the cosine function is an essential part of trigonometry. The cosine function is a periodic function that repeats its values in regular intervals or periods. For the standard cosine function, these intervals span over a period of \(2\pi\). When a cosine function is modified, such as in the equation \( y = 3 - 4 \cos x \), it involves transformations. These modifications may include:

• Amplitude change: The factor of \(-4\) before \( \cos x \) signifies a stretch in the graph's height by a factor of 4 and a reflection across the x-axis, making the peaks and troughs more pronounced and inverted compared to the standard \( \cos x \).
• Vertical Shift: The '+3' shifts the whole graph upwards by 3 units, making the baseline around which the function oscillates change from \( y = 0 \) to \( y = 3 \).

By understanding these modifications, you can predict how these affects change the typical cosine curve.

Graphing Trigonometric Functions

Graphing trigonometric functions like the cosine function involves identifying key characteristics such as amplitude, period, phase shift, and vertical shift. For the given function \( y = 3 - 4 \cos x \), these aspects showcase how the function behaves visually.To graph one cycle:

• Amplitude and Inversion: The amplitude is 4, due to \(|-4|\), and the negative sign before the 4 causes the graph to flip and stretch vertically. This means our peaks and troughs are 8 units apart, from the highest point to the lowest point, centered at the vertical shift line.
• Period: Here, the function's period is \(2\pi\), which means the graph will complete one full cycle in this interval. Critical points occur where the cosine achieves a value of 1, -1, and 0. This results in peaks and valleys at intervals of \(\pi\).
• Vertical Shift: A shift of 3 units upward means the oscillation is centered around \(y = 3\) instead of \(y = 0\), resulting in a range from -1 to 7.

By marking these points, one can sketch a smooth, consistent wave illustrating the cosine's cyclic nature.

Range of Functions

The range of a function defines the set of possible output values it can take. In trigonometric functions, range is determined by its amplitude and any vertical shifts present in the function.For our exercise function \(y = 3 - 4 \cos x\):

• Amplitude's Role: The amplitude of \(4\) means that the cosine component can at most contribute to the vertical movement by 4 units, which combined with the vertical shift, helps fix the highest and lowest points the graph can reach.
• Calculating Maximum and Minimum: The vertical shift of 3 added to the minimum and maximum contributions of the cosine part, which ranges naturally from -1 to 1, results in absolute bounds becoming \(7\) at maximum and \(-1\)at minimum.

Thus, the function oscillates between these calculated bounds and its formal mathematical range is noted as \([-1, 7]\). This recognizes all values the function can achieve within any cycle.