Problem 36 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Describe and correct the error in determining the point where the maximum value of the function \(y=2 \sin \left(x-\frac{\pi}{2}\right)\) occurs.

Step-by-Step Solution

Answer

The maximum value of the function \(y=2\sin\left(x-\frac{\pi}{2}\right)\) occurs at \(x=\pi\). The error in determining the maximum value of the function could have been forgetting or incorrectly applying the phase shift of the sine function.

Step 1: Identify the function modification

We first need to identify the type of the sine function we have. Here we observe that the sine function is translated and scaled. The function \(y=a \sin (b(x-c))\) is a sine function with amplitude \(a\), period \(2\pi/b\), and phase shift \(c\). In our case, \(a=2\), \(b=1\), and \(c=\frac{\pi}{2}\).

Step 2: Determine the phase shift

The phase shift indicates how the function is shifted horizontally. We can see that the phase shift is \(\frac{\pi}{2}\) to the right. This means the maximum point of the function, which for a standard sine function is at \(\pi/2\) is now shifted to the right by \(\pi/2\). Therefore, the maximum point of our function would be at \(x=\pi\).

Step 3: Verify the result

The correctness of the result can be verified by simple substitution. Substitute \(x=\pi\) into the original function we get \(y= 2\sin(\pi - \frac{\pi}{2}) = 2\sin(\frac{\pi}{2}) = 2 * 1 = 2\). Since 2 is the maximum value of the function as the amplitude is 2, \(pi\) is indeed the x-coordinate where the maximum value is obtained.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions, commonly used to describe periodic phenomena such as sound waves and light waves. In its simplest form, it is expressed as \( y = \sin(x) \). This function oscillates between -1 and 1, forming a smooth wave-like curve.

When dealing with transformations of the sine function, we encounter equations of the form \( y = a \sin(b(x - c)) \). Here, each parameter plays an important role:

• \( a \) affects the amplitude, determining how "tall" or "short" the waves are.
• \( b \) impacts the period, or the distance over which the wave pattern repeats.
• \( c \) refers to the phase shift, which moves the wave horizontally on the graph.

Understanding these components helps in predicting the behavior of the function and in graphing it accurately.

Amplitude

Amplitude in a sine function dictates how high and low the function goes. It represents half the distance between the maximum and minimum values of the function.

For a function like \( y = a \sin(x) \), the amplitude is given by \( |a| \).
So, in our specific function \( y = 2 \sin(x - \frac{\pi}{2}) \), the amplitude is 2. This tells us that the waves of the sine function will reach up to 2 and down to -2.

• If \( a \) is positive, the wave pattern begins rising from the origin.
• If \( a \) is negative, the wave pattern starts by dipping below the origin, essentially flipping over the x-axis.

Amplitude is an important property, especially when interpreting real-life data, like sound levels or degrees of motion, which are often described using sine functions.

Phase Shift

A phase shift refers to the horizontal movement of a sine wave on a graph. It's determined by the \( c \) in the function equation \( y = a \sin(b(x-c)) \).

The phase shift can cause the wave to move to the right or left based on the value of \( c \).

• When \( c > 0 \), the wave shifts to the right.
• When \( c < 0 \), it moves to the left.

In our example, \( y = 2 \sin(x - \frac{\pi}{2}) \), the phase shift is \( \frac{\pi}{2} \) to the right. This means the standard starting point of the sine function is adjusted by \( \frac{\pi}{2} \), altering where the wave's peaks and troughs occur.

Understanding phase shifts is crucial, especially in applications like signal processing, where the timing of wave features is important.

Maximum Value

The maximum value of a sine function is determined by the sum of its amplitude and vertical translation if any.

For a typical sine function like \( y = a \sin(b(x-c)) + d \), the maximum value is \( d + |a| \). In scenarios without vertical shifts, \( d = 0 \), simplifying the maximum value to just the amplitude \( |a| \).

In our case, \(y=2 \sin(x - \frac{\pi}{2})\), the maximum value is precisely the amplitude, \( 2 \).

Finding maximum values is particularly useful:

• In understanding cycles in natural processes, such as tides or day and night length.
• For accurately managing systems that rely on peak values, like oscillating circuits in electronics.

Knowing how to pinpoint and verify maximum values is essential in many scientific and engineering disciplines.