Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Find the (a) amplitude, (b) period, (c) phase shift (if any). (d) vertical translation (if any), and (e) range of each finction. Then graph the function over at least one period. $$y=-3+2 \sin \left(x+\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

Amplitude: 2, Period: \(2\pi\), Phase Shift: \(\frac{\pi}{2}\) left, Vertical Translation: 3 down, Range: \([-5, -1]\).

Step 1: Identify the Function Form

The given function is \(y = -3 + 2 \sin(x + \frac{\pi}{2})\). This can be seen as \(y = a + b \sin(c(x - d))\) where \(a = -3\), \(b = 2\), \(c = 1\), and \(d = -\frac{\pi}{2}\).

Step 2: Find the Amplitude

The amplitude of a sine function is given by the absolute value of the coefficient of \(\sin\), which is \(|b|\). For this function, the amplitude is \(|2| = 2\).

Step 3: Determine the Period

The period of the sine function \(\sin(cx)\) is given by \(\frac{2\pi}{c}\). Since \(c = 1\) in the equation, the period is \(\frac{2\pi}{1} = 2\pi\).

Step 4: Calculate the Phase Shift

The phase shift can be found by solving \(c(x - d) = x + \frac{\pi}{2}\), which is \(-d\). Thus, the phase shift is \(-\left(-\frac{\pi}{2}\right) = \frac{\pi}{2}\) to the left.

Step 5: Identify the Vertical Translation

The vertical translation is given by the constant \(a\) added to the entire function. In this case, it's \(-3\), meaning the graph is shifted 3 units down.

Step 6: Determine the Range

The typical range of \(\sin(x)\) is \([-1, 1]\). With amplitude \(2\) and vertical shift \(-3\), the range becomes \([-3 - 2, -3 + 2] = [-5, -1]\).

Step 7: Graph the Function

To graph the function over one period: \(-3 + 2 \sin(x + \frac{\pi}{2})\), start from \(x = -\frac{\pi}{2} \) to \(x = \frac{3\pi}{2}\) to cover a full period \(2\pi\). Then plot the transformed sine wave with amplitude 2, phase shift left \(\frac{\pi}{2}\), and vertical shift down 3.

Key Concepts

Amplitude

The amplitude of a trigonometric function describes the height of the wave from its central axis to the maximum or minimum point. In the function given, which is \(y = -3 + 2 \sin(x + \frac{\pi}{2})\), the amplitude is determined by the absolute value of the coefficient in front of the sine function. Here, that coefficient is 2.
The amplitude is therefore \(|2| = 2\). This means that from the function's central position, the wave will reach up to 2 units above and 2 units below.

• The amplitude helps in understanding how tall the peaks and how low the troughs of the wave are.
• Shorter amplitudes result in flatter waves, while larger amplitudes produce taller waves.

Phase Shift

The phase shift of a trigonometric function refers to the horizontal displacement of the wave from its usual starting point. In the function \(y = -3 + 2 \sin(x + \frac{\pi}{2})\), the phase shift is calculated by considering the term inside the sine function.
The formula typically used is \(x = x - d\), where \(d\) is the horizontal shift. Here, our function looks like \(\sin\left(x + \frac{\pi}{2}\right)\), indicating a shift.
However, because of the addition sign, a shift of \(-\frac{\pi}{2}\) means the function actually moves left by \(\frac{\pi}{2}\).

• Positive numbers inside the function mean shifting to the left.
• Negative numbers inside suggest shifting to the right.

Vertical Translation

Vertical translation moves the graph of the function up or down along the y-axis without affecting its shape. In our function \(y = -3 + 2 \sin(x + \frac{\pi}{2})\), this shift is determined by the constant \(-3\), added to the sine function.
This means the entire wave is moved downwards by 3 units.

• The middle of the sine wave, normally at zero, gets moved to -3 on the y-axis.
• This affects the range of the function, as the entire wave now covers a lower set of values.

Period

The period of a sine function is the length of one complete cycle of the wave. It helps us understand how frequently the cycle repeats itself horizontally.
For the sine function \(\sin(cx)\), the period is calculated using the formula \(\frac{2\pi}{c}\).
In our example \(y = -3 + 2 \sin(x + \frac{\pi}{2})\), since \(c = 1\), the period of the function is \(\frac{2\pi}{1} = 2\pi\).

• A shorter period would mean the wave cycles faster over a shorter length.
• A longer period stretches the cycle over a greater horizontal distance.

Range

The range of a trigonometric function indicates the set of y-values the function can take. For the standard sine function \(\sin(x)\), the range is typically \([-1, 1]\). However, transformations can alter this range.
For the function \(y = -3 + 2 \sin(x + \frac{\pi}{2})\):

• The amplitude of 2 will stretch the minimum and maximum values from -1 and 1 to -2 and 2.
• The vertical translation of -3 shifts the range down.): \([-2 + (-3), 2 + (-3)] = [-5, -1]\).

Thus, the modified range of this function is \([-5, -1]\). This altered range accounts for various transformations applied to the sine wave, indicating new maximum and minimum y-values.