Problem 64 - Precalculus

Question

In Exercises \(61-66,\) sketch the graph of the function over the indicated interval. $$y=-\frac{1}{2}+\frac{1}{2} \cos \left(\frac{1}{2} x+\frac{\pi}{4}\right),\left[-\frac{9 \pi}{2}, \frac{7 \pi}{2}\right]$$

Step-by-Step Solution

Answer

The graph is a cosine wave shifted downward by 1/2 unit, scaled to amplitude 1/2, and with a period of 4π covering [-9π/2, 7π/2].

Step 1: Identify the Function Type

The given function is a transformed cosine function. It's important to recognize the general form as it helps in understanding how the graph will be shifted or stretched.

Step 2: Understand the Function Transformations

The function is written as: \[ y = -\frac{1}{2} + \frac{1}{2} \cos \left(\frac{1}{2}x + \frac{\pi}{4}\right) \]This indicates:- Vertical Shift: -1/2 units (downwards)- Amplitude: 1/2 (the height from the center of the graph to a peak)- Horizontal Shift: -\( \frac{\pi}{4} \) units (left)- Period: The period is \( \frac{2\pi}{\frac{1}{2}} = 4\pi \)

Step 3: Determine Key Points

For one full period, key points will occur at regular intervals of \( \frac{\text{Period}}{4} = \pi \). These points are: 1. Starting point at \( x = -\frac{\pi}{4} \)2. Midpoint of the drop at \( x = \frac{3\pi}{4} \)3. Bottom at \( x = \frac{7\pi}{4} \)4. Rising midpoint at \( x = \frac{11\pi}{4} \)5. Completion of a cycle at \( x = \frac{15\pi}{4} \)

Step 4: Scale and Translate these Points

Calculate the y-values using the cosine function:- At \( x = -\frac{\pi}{4} \), \( y = -\frac{1}{2} + \frac{1}{2} \cos(0) = 0 \)- At \( x = \frac{3\pi}{4} \), \( y = -\frac{1}{2} + \frac{1}{2} \cos(\frac{\pi}{2}) = -\frac{1}{2} \)- At \( x = \frac{7\pi}{4} \), \( y = -\frac{1}{2} + \frac{1}{2} \cos(\pi) = -1 \)- At \( x = \frac{11\pi}{4} \), \( y = -\frac{1}{2} + \frac{1}{2} \cos(\frac{3\pi}{2}) = -\frac{1}{2} \)- At \( x = \frac{15\pi}{4} \), \( y = -\frac{1}{2} + \frac{1}{2} \cos(2\pi) = 0 \)

Step 5: Sketch the Graph

Using the points calculated and considering the transformations, sketch the cosine wave over the interval \([-\frac{9\pi}{2}, \frac{7\pi}{2}]\). The cosine wave will repeat every \(4\pi\), so graph two complete cycles to cover this interval, starting from \( x = -\frac{9\pi}{2} \). Ensure each cycle mirrors the points stated above.

Key Concepts

Understanding the Cosine Function

The cosine function is one of the fundamental trigonometric functions. It is often represented in the standard form as \( y = \cos{x} \). This function oscillates between +1 and -1, forming a wave-like pattern when graphed.

Key characteristics of the cosine function include:

• The function's graph is a smooth continuous wave, repeating every \( 2\pi \) radians, which is known as its period.
• It begins at its maximum value of 1 when \( x = 0 \).
• It crosses the x-axis at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \), continuing this pattern.
• The minimum value of -1 is achieved at \( x = \pi \).

Understanding the basic form of the cosine function is critical as it serves as the foundation for learning about its transformations, which involve altering the amplitude, period, phase shift, and vertical shift.

Exploring Function Transformation

Function transformation develops from altering the basic formula of a trigonometric function like cosine to produce a variety of graph shapes. These transformations involve shifting, stretching, or compressing the graph in different ways, thereby changing its appearance.

For the function \( y = -\frac{1}{2} + \frac{1}{2}\cos\left(\frac{1}{2}x + \frac{\pi}{4}\right) \), the key transformations include:

• Vertical Shift: The term \(-\frac{1}{2}\) shifts the entire graph downward by 0.5 units.
• Amplitude: The coefficient \( \frac{1}{2} \) reduces the graph's height, meaning it oscillates between \(-1\) and \(0\) after the vertical shift.
• Horizontal Shift (Phase Shift): The term \(\frac{\pi}{4}\) in the angle translates the graph to the left by \(\frac{\pi}{4}\).
• Period Change: The factor \( \frac{1}{2} \) inside the cosine function indicates that the period becomes larger, doubling to \( 4\pi \).

These transformations allow mathematicians to modify the graph to model different phenomena or to match specific data more closely.

Calculating the Period of Trigonometric Function

The period of a trigonometric function is the length over which the function starts repeating the same wave pattern. It's an essential property that defines how often the wave will complete a full cycle.

For the basic cosine function \( y = \cos{x} \), the period is \( 2\pi \), meaning every \( 2\pi \) units along the x-axis, the function will start repeating its wave. The modified period is calculated by changing the frequency within the cosine function itself:

• If the function is \( y = \cos(bx) \), the period is \( \frac{2\pi}{b} \).

In our specific example \( y = -\frac{1}{2} + \frac{1}{2}\cos\left(\frac{1}{2}x + \frac{\pi}{4}\right) \), the period is calculated as:
\[\text{Period} = \frac{2\pi}{\frac{1}{2}} = 4\pi\] This increased period indicates the wave will elongate, repeating itself more slowly over the x-axis.

Interpreting Amplitude in the Graph

Amplitude is a measure of how far the graph of the cosine function stretches vertically. More specifically, it defines the height from the center of the wave to its peak. The standard cosine function has an amplitude of 1, as it peaks at 1 and troughs at -1 from a midline of 0.

In trigonometric functions, the amplitude is determined by the coefficient multiplied by the cosine term. For example, in \( y = A\cos{x} \), the amplitude is \( |A| \).

• In the provided function \( y = -\frac{1}{2} + \frac{1}{2}\cos\left(\frac{1}{2}x + \frac{\pi}{4}\right) \), the amplitude is \( \frac{1}{2} \).

This value indicates the graph will peak at \( -\frac{1}{2} + \frac{1}{2} = 0 \) and reach a minimum at \( -\frac{1}{2} - \frac{1}{2} = -1 \), showcasing a smaller vertical stretch compared to the untransformed cosine function.

Understanding Phase Shift

Phase shift indicates the amount by which a trigonometric graph is shifted horizontally from its standard position. This shift determines where the wave starts along the x-axis.

The phase shift in functions like \( y = \cos(bx + c) \) is calculated as \( -\frac{c}{b} \), reversing the sign convention typically seen in linear equations.

• For the example function \( y = -\frac{1}{2} + \frac{1}{2}\cos\left(\frac{1}{2}x + \frac{\pi}{4}\right) \), we calculate the phase shift as:
  \[\text{Phase Shift} = -\frac{\frac{\pi}{4}}{\frac{1}{2}} = -\frac{\pi}{2}\]

This negative value indicates the graph is shifted \( \frac{\pi}{2} \) units to the left, meaning the wave peaks and valleys occur earlier compared to the unshifted cosine graph. Understanding this concept allows you to predict the starting point and alignment of the graph.