Question

Write the equation of the sine function in the form \(y=a \sin b(x-c)+d\) given its characteristics. a) amplitude \(4,\) period \(\pi,\) phase shift \(\frac{\pi}{2}\) to the right, vertical displacement 6 units down b) amplitude 0.5, period \(4 \pi,\) phase shift \(\frac{\pi}{6}\) to the left, vertical displacement 1 unit up c) amplitude \(\frac{3}{4},\) period \(720^{\circ},\) no phase shift, vertical displacement 5 units down

Short answer

Part (a): y = 4 \sin 2(x - \frac{\pi}{2}) - 6; Part (b): y = 0.5 \sin \frac{1}{2}(x + \frac{\pi}{6}) + 1; Part (c): y = \frac{3}{4} \sin \frac{1}{2}x - 5.

Step-by-step solution

Understand the General Form of the Sine Function

The general form of a sine function is given by \[ y = a \ \sin b(x-c) + d \] where:- \(a\) is the amplitude- \(\frac{2\pi}{b}\) is the period- \(c\) is the phase shift- \(d\) is the vertical displacement.

Plug in the Values for Part (a)

For part (a):- Amplitude \(a = 4\)- Period \(\pi\) implies \( b = \frac{2\pi}{\pi} = 2\)- Phase shift of \(\frac{\pi}{2}\) to the right implies \(c = \frac{\pi}{2}\)- Vertical displacement 6 units down implies \(d = -6\)Therefore, the equation is \[ y = 4 \ \sin 2(x - \frac{\pi}{2}) - 6 \]

Plug in the Values for Part (b)

For part (b):- Amplitude \(a = 0.5\)- Period \(4\pi\) implies \( b = \frac{2\pi}{4\pi} = \frac{1}{2}\)- Phase shift of \(\frac{\pi}{6}\) to the left implies \(c = -\frac{\pi}{6}\)- Vertical displacement 1 unit up implies \(d = 1\)Therefore, the equation is \[ y = 0.5 \ \sin \frac{1}{2}(x + \frac{\pi}{6}) + 1 \]

Plug in the Values for Part (c)

For part (c):- Amplitude \(a = \frac{3}{4}\)- Period \(720^{\circ}\) implies \( b = \frac{360^{\circ}}{720^{\circ}} = \frac{1}{2}\) (converted to degrees)- No phase shift implies \(c = 0\)- Vertical displacement 5 units down implies \(d = -5\)Therefore, the equation is \[ y = \frac{3}{4} \ \sin \frac{1}{2}x - 5 \]

Key concepts

Amplitude and Its Role in Sine Functions

The amplitude of a sine function determines the height of its peaks and the depth of its troughs. It is represented by the variable \(a\) in the function \(y = a \sin b(x-c) + d\).
Amplitude is a measure of how much the function's value deviates from the centerline. For example:

• If the amplitude is 4, the sine wave will reach values of 4 and -4 at its highest and lowest points, respectively.
• If the amplitude is 0.5, the sine wave will only reach values of 0.5 and -0.5 at its extremes.

This means that larger amplitudes result in taller waves, while smaller amplitudes result in more compressed waves.
Amplitudes are never negative; however, a negative amplitude would reflect the sine wave across the horizontal axis. Always remember:

• A larger amplitude means a more pronounced wave.
• A smaller amplitude means a less pronounced wave.

Understanding the Period of Sine Functions

The period of a sine function defines how long it takes for the sine wave to complete one full cycle. In the general sine equation \(y = a \sin b(x-c) + d\), the period is calculated as \(\frac{2\pi}{b}\).
The period influences the width of the cycles. For instance:

• A period of \(\pi\) implies that the sine wave completes a full cycle within a horizontal interval of \(\pi\).
• A period of \(4\pi\) means the wave stretches out to complete one cycle over \(4\pi\). This results in a more spread-out wave.

This can be visualized as:

• A smaller period results in a more frequent or tightly packed wave.
• A larger period results in a less frequent or more stretched-out wave.

The conversion between angles and radians often comes into play, as seen in part (c) where the period in degrees is converted for consistency.
Never forget:

• The 'b' value in the sine equation stretches or compresses the wave by adjusting the period.

Phase Shift of Sine Functions

The phase shift of a sine function determines where the sine wave begins along the x-axis. This is given by the variable \(c\) in the general form \(y = a \sin b(x-c) + d\).
Phase shift essentially 'moves' the wave left or right:

• A positive phase shift (\(c\) positive) moves the wave to the right.
• A negative phase shift (\(c\) negative) moves the wave to the left.

For example:

• In part (a), a phase shift of \(\frac{\pi}{2}\) to the right means the wave starts at \(x = \frac{\pi}{2}\).
• In part (b), a phase shift of \(\frac{\pi}{6}\) to the left means the wave starts at \(x = -\frac{\pi}{6}\).

Understanding phase shifts is crucial for graphing sine functions accurately and ensuring the wave starts from the intended point along the x-axis.
Key points to remember:

• Phase shift moves the wave along the x-axis without changing its amplitude or period.

Vertical Displacement in Sine Functions

The vertical displacement of a sine function is represented by the variable \(d\) in the equation \(y = a \sin b(x-c) + d\). This shifts the entire sine wave up or down along the y-axis.
Vertical displacement is responsible for where the midline of the wave is positioned:

• A negative vertical displacement moves the wave downward.
• A positive vertical displacement moves the wave upward.

In practice:

• A vertical displacement of -6, as in part (a), shifts the wave 6 units downward.
• A vertical displacement of 1, as in part (b), lifts the wave 1 unit upward.

This affects the overall positioning of all the peaks and troughs, but not their relative distances.
Always bear in mind:

• Vertical displacement adjusts the baseline or midline of the sinusoidal wave, not the pattern of the wave itself.