Problem 31 - Algebra and Trigonometry

Question

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=-4 \sin 2\left(x+\frac{\pi}{2}\right) $$

Step-by-Step Solution

Answer

Amplitude is 4; period is \(\pi\); phase shift is \(-\frac{\pi}{2}\).

Step 1: Identify the amplitude

The amplitude of a sine function in the form \(y = a \sin(b(x - c))\) is the absolute value of \(a\). In the given function \(y = -4 \sin(2(x + \frac{\pi}{2}))\), the coefficient \(a = -4\). Therefore, the amplitude is \(|a| = |-4| = 4\).

Step 2: Determine the period

The period of a sine function is given by the formula \(\frac{2\pi}{b}\), where \(b\) is the coefficient of \(x\) inside the sine function. Here, \(b = 2\). So, the period is \(\frac{2\pi}{2} = \pi\).

Step 3: Calculate the phase shift

The phase shift is determined using \(c\) in the formula \(y = a \sin(b(x - c))\). The equation is \(y = -4 \sin(2(x + \frac{\pi}{2}))\), which can be rewritten as \(y = -4 \sin(2(x - (-\frac{\pi}{2})))\). Therefore, the phase shift is \(-\frac{\pi}{2}\), indicating a left shift of \(\frac{\pi}{2}\).

Step 4: Graph one complete period

To graph one complete period of the function, plot the function starting from the phase shift \(-\frac{\pi}{2}\) and covering a range of \(\pi\). The key points in one period of \(\sin\) (starting at 0 and going to \(2\pi\) in the usual sine curve) will now occur at \(-\frac{\pi}{2}\), \(-\frac{\pi}{2} + \frac{\pi}{4}\), \(-\frac{\pi}{2} + \frac{\pi}{2}\), \(-\frac{\pi}{2} + \frac{3\pi}{4}\), and \(-\frac{\pi}{2} + \pi\). Use these points to plot the curve reflecting the amplitude of 4 and picking values accordingly.

Key Concepts

Amplitude

The amplitude of a trigonometric function like the sine function refers to its maximum height from the centerline, or the line y=0 in the most basic sine functions. In simple terms, it's how far up and down the curve travels from the middle of its wave. The standard formula for a sine function is represented as \(y = a \sin(b(x - c))\), where \(a\) controls the amplitude. Here, the given function is \(y = -4 \sin(2(x + \frac{\pi}{2}))\). The amplitude is the absolute value of \(a\), which means we disregard any negative sign present.

• Given \(a = -4\), the amplitude is \(|-4| = 4\).
• This indicates that the graph of the sine function will oscillate 4 units above and 4 units below the x-axis.

Understanding amplitude helps determine how "tall" your graph will look as it varies above and below the centerline.

Period

The period of a sine function is the distance over which the graph of the function repeats itself. For the basic sine curve \(y = \sin(x)\), the period is standardly \(2\pi\), meaning after \(2\pi\) units the wave starts repeating. The formula to find the period for a transformed sine function \(y = a \sin(b(x - c))\) is calculated as \(\frac{2\pi}{b}\). In our function \(y = -4 \sin(2(x + \frac{\pi}{2}))\), the \(b\) value is 2.

• Compute the period: \(\frac{2\pi}{2} = \pi\).
• This tells us the wave will repeat every \(\pi\) units, compressing the standard sine wave to half its length.

Knowing the period is crucial for graphing as it tells you the horizontal length required to capture one complete cycle of the sine wave.

Phase Shift

The phase shift of a sine function is the horizontal translation or movement of the graph along the x-axis. It alters where the sine wave starts on the graph. Using the expression \(y = a \sin(b(x - c))\), the phase shift is determined by the value \(c\).In the function given, \(y = -4 \sin(2(x + \frac{\pi}{2}))\), it can be set as \(y = -4 \sin(2(x - (-\frac{\pi}{2})))\).

• This reveals a phase shift of \(-\frac{\pi}{2}\), meaning the graph is shifted to the left by that distance.

Understanding phase shift helps predict the starting point of the wave and how it aligns with standard key positions like peaks and troughs.

Graphing Sine Functions

Graphing a sine function can be simplified into steps once you know its amplitude, period, and phase shift. These elements allow you to determine key points on the graph and thus shape the sine wave. The standard form of a sine function, \(y = a \sin(b(x - c))\), is transformed to create different sine curves.In our equation \(y = -4 \sin(2(x + \frac{\pi}{2}))\):

• The amplitude is 4, so the graph will extend 4 units above and below the centerline.
• The period is \(\pi\), you plot each cycle across this length.
• With a phase shift of \(-\frac{\pi}{2}\), you start plotting your cycle from that point on the x-axis.

To graph, you'd begin plotting at the phase shift, marking points at critical values (like peak, zero, trough, zero again) within the period. For instance, one complete cycle begins at \(-\frac{\pi}{2}\), ends at \(\pi - \frac{\pi}{2}\), and maps out a smooth wave through these guiding points.