Problem 49 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Which of the following is a point where the maximum value of the graph of \(y=-4 \cos \left(x-\frac{\pi}{2}\right)\) occurs? (A) \(\left(-\frac{\pi}{2}, 4\right)\) (B) \(\left(\frac{\pi}{2}, 4\right)\) (C) \((0,4)\) (D) \((\pi, 4)\)

Step-by-Step Solution

Answer

None of the provided options accurately represents the location where the maximum value of the function \(y=-4 \cos \left(x-\frac{\pi}{2}\right)\) occurs. The correct answer should be \(\left(\frac{\pi}{2}, -4\right)\). Therefore, the question might contain a typo and requires clarification.

Step 1: Analyze the function

Firstly, it is key to understand the properties of the function used in this exercise. The given function is \(y=-4 \cos \left(x-\frac{\pi}{2}\right)\), a variation of the cosine function. The number -4 is the amplitude, which is the peak deviation of the function from zero. This means, the maximum value this function can reach is -4, not 4, because the function is inverted due to the negative sign.

Step 2: Identify the phase shift

Phase shift refers to the horizontal movement of the wave. In this case, we have \(x-\frac{\pi}{2}\) inside the cosine function so the function is shifted \(\frac{\pi}{2}\) units to the right. Keeping in mind that the cosine function begins with a maximum at x=0, the maximum for this function will be at \(x=\frac{\pi}{2}\). However, remember that the value will be -4 not 4 as the function is inverted.

Step 3: Identify the potential solution

Now, evaluate the provided options. Essentially, we are looking for a point where x has the value \(\frac{\pi}{2}\) and y has the value -4. None of the options have the y-value -4. This indicates that none of the provided options will be an accurate representation of the location where the maximum value of the given function occurs.

Key Concepts

Cosine Function

The cosine function is one of the primary trigonometric functions, and it's crucial when studying periodic behaviors in math and science. This function is often symbolized as \( \cos(x) \) where \( x \) is the angle in radians. The standard cosine function starts at its maximum value of 1 when \( x = 0 \). Its values gradually decrease to -1 as it approaches \( \pi \) radians, then returns to 1 at \( 2\pi \) radians, forming a wave pattern on a graph.
The cosine function is well-known for its wave-like symmetrical graph that continues forever in both directions. It's often used to model cyclic behaviors, such as sound waves or seasonal weather patterns, because of its predictable oscillations.

Amplitude

Amplitude in trigonometry refers to the height from the center line of the wave to its peak. In simpler terms, it's how "tall" the wave is. For a function like \( y = -4 \cos(x) \), the amplitude is 4, even with the negative sign. However, the "-" in front of the cosine function indicates that the graph is flipped vertically. As a result, instead of the graph oscillating between 0 and 4, it oscillates between 0 and -4. This negative amplitude makes sure the wave pattern of the cosine function reflects over the horizontal axis.
You often see amplitude when trying to determine how much variation there is from the mean value in a periodic function. It's directly correlated with the energy of the wave when interpreted in physical contexts.

Phase Shift

The phase shift in a trigonometric function describes its horizontal translation along the x-axis. It's essentially the amount by which the graph of a function is shifted horizontally. Looking at the function \( y = -4 \cos(x - \frac{\pi}{2}) \), the term \( (x - \frac{\pi}{2}) \) indicates a phase shift. This means that the graph of the basic cosine function is shifted \( \frac{\pi}{2} \) units to the right.
Understanding phase shifts is fundamental in identifying how a wave will appear in a different context. It's a vital tool in signal processing, where knowing the shift allows for the proper alignment of waves for interpretation or transmission.

Graph Analysis

Analyzing the graph of a trigonometric function like the cosine function involves a few critical steps: identifying amplitude, frequency, phase shifts, and vertical shifts. With the function \( y = -4 \cos(x - \frac{\pi}{2}) \), graph analysis begins with recognizing the amplitude of 4, which is essentially the peak value the wave can achieve. However, due to the negative sign, this peak is actually -4, and this translates to the entire wave being flipped upside down.
Next, the phase shift \( \frac{\pi}{2} \) indicates the graph moves to the right by \( \frac{\pi}{2} \) units. This is crucial in determining where the graph's maximum points occur. Originally, for \( \cos(x) \), this would be at 0 but shifts to \( \frac{\pi}{2} \) due to the phase shift.

• Amplitude affects the height
• Phase shift affects horizontal placement
• Together, they determine the wave's appearance

This kind of graph analysis is a foundation of understanding trigonometric functions and their applications in real-world scenarios such as sound engineering or electrical circuits.