Problem 1 - Precalculus

Question

In Exercises \(1-12,\) graph one cycle of the given function. State the period, amplitude, phase shift and vertical shift of the function. \(y=3 \sin (x)\)

Step-by-Step Solution

Answer

Period: \(2\pi\), Amplitude: 3, Phase Shift: 0, Vertical Shift: 0.

Step 1: Identify the Standard Form of the Sine Function

The standard form of the sine function is given by \( y = A \sin(Bx - C) + D \). From this, we can identify the amplitude \(A\), the frequency \(B\), the phase shift from \(C\), and the vertical shift from \(D\).

Step 2: Determine the Amplitude

In the equation \( y = 3 \sin(x) \), the coefficient \(A\) represents the amplitude of the sine wave. Therefore, the amplitude is \(A = 3\).

Step 3: Find the Period of the Function

The period of a sine function \( y = \, A \sin(Bx) \, \) is calculated as \( \frac{2\pi}{B} \). In this function, \(B = 1\), so the period is \( \frac{2\pi}{1} = 2\pi \).

Step 4: Identify the Phase Shift

The phase shift \( \phi \) of the function is calculated using the expression \( \frac{C}{B} \). Since \(C = 0\) and \(B = 1\), we have \( \phi = \frac{0}{1} = 0 \). Thus, there is no phase shift in this function.

Step 5: Determine the Vertical Shift

The vertical shift \(D\) in the standard form is determined by the constant added to the function. Here, \(D = 0\), therefore, there is no vertical shift.

Step 6: Graph One Cycle of the Function

Plot the sine wave over one period, \([0, 2\pi]\), with an amplitude of 3, no phase shift, and no vertical shift. The function reaches its maximum at \(y = 3\), its minimum at \(y = -3\), and passes through the origin \( (0, 0) \). Additional key points are \( (\pi/2, 3), \pi, 0), (3\pi/2, -3), \) and \( (2\pi, 0) \).

Key Concepts

Amplitude

The amplitude of a sine function is one of its most distinct characteristics. It tells us how high and low the function reaches. In the standard sine form, given by \[ y = A \sin(Bx - C) + D \]the amplitude is denoted by the constant \( A \). This value indicates the peak deviation from the function's central axis.

In the function \( y = 3 \sin(x) \), the amplitude is \( 3 \). This means the sine wave rises to a maximum of \( 3 \) and dips to a minimum of \( -3 \).

• The larger the value of \( A \), the taller the peaks and the deeper the troughs.
• If \( A \) is negative, it reflects the graph across the x-axis.

It's crucial to remember that amplitude is always taken as a positive number, reflecting the maximum displacement both above and below the midline.

Period

The period of a sine function describes how long it takes for the function to complete one full cycle, repeating its pattern. Understanding the period helps visualize how frequent the wave oscillates.To find the period in the expression of the form \( y = A \sin(Bx - C) + D \), use the formula:\[ \text{Period} = \frac{2\pi}{B} \]In the case of \( y = 3 \sin(x) \), since \( B = 1 \), the period is:\[ \frac{2\pi}{1} = 2\pi \]This indicates the function powerfully completes a cycle spanning \( 2\pi \) units along the x-axis.

• As \( B \) increases, the wave compresses, meaning shorter periods.
• Conversely, reducing \( B \) results in a stretched wave, increasing the period.

Phase Shift

The phase shift indicates a horizontal shift from the standard position, showing if the wave is moved left or right on the graph. In mathematical terms, it's driven by the \( C \) and \( B \) values in the formula \( y = A \sin(Bx - C) + D \).We calculate phase shift using:\[ \phi = \frac{C}{B} \]For the function \( y = 3 \sin(x) \), \( C = 0 \) and \( B = 1 \), making the phase shift:\[ \phi = \frac{0}{1} = 0 \]This means there is no phase shift, and the sine wave retains its basic starting position without sliding horizontally on the graph.

• A positive phase shift moves the graph to the right.
• A negative phase shift moves it to the left.

Vertical Shift

The vertical shift represents a movement of the wave along the y-axis, either upwards or downwards. It's determined by the constant \( D \) in the sine equation:\[ y = A \sin(Bx - C) + D \]In the function \( y = 3 \sin(x) \), \( D \) is \( 0 \), signifying there's no vertical shift. The graph oscillates around the horizontal axis at \( y = 0 \), which acts as the midline.

• If \( D \) is positive, the entire graph shifts upwards by \( D \) units.
• A negative \( D \) value moves the graph downwards by \( |D| \) units.

Comprehending vertical shifts is vital for adjusting and recognizing how sine waves move relative to the x-axis. When there’s no shift, like in this function, the central axis remains at the y-coordinate of \( 0 \), maintaining symmetry around y.