Problem 11 - Calculus Concepts

Question

For each of the functions, state the amplitude, period, average value, and horizontal shift. \(\quad y(x)=\sin x\)

Step-by-Step Solution

Answer

Amplitude: 1, Period: \(2\pi\), Average Value: 0, Horizontal Shift: 0.

Step 1: Identify the Amplitude

The amplitude of a sine function in the form \( y = a \sin(bx + c) + d \) is given by the absolute value of \( a \). For \( y(x) = \sin x \), the amplitude \( |a| \) is \( 1 \).

Step 2: Determine the Period

The period of a sine function is given by \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) in \( y = a \sin(bx + c) + d \). For \( y(x) = \sin x \), \( b = 1 \), so the period is \( \frac{2\pi}{1} = 2\pi \).

Step 3: Find the Average Value

The average value of a sine function over one period is \( d \). Since \( d = 0 \) for \( y(x) = \sin x \), the average value is \( 0 \).

Step 4: Determine the Horizontal Shift

The horizontal shift is determined by \( \frac{-c}{b} \). For \( y(x) = \sin(x) \), there is no \( c \) or horizontal shift, so \( \frac{-0}{1} = 0 \).

Key Concepts

Amplitude of Sine Functions

The amplitude of a sine function represents the maximum height of the wave from its average, or middle, value. It's how tall the wave peaks. In general, if you have a function of the form \( y = a \sin(bx + c) + d \), the amplitude is determined by \(|a|\), the absolute value of the coefficient \(a\).
For instance, in the most basic sine function \( y(x) = \sin x \), the amplitude is \(1\). This means the wave reaches a maximum of \(1\) unit above and a minimum of \(-1\) unit below the center line of the wave (which is at \(y=d\)).
The amplitude tells us about the strength or intensity of the oscillation in the wave pattern. So, remember:

• The larger the amplitude, the taller the wave.
• The amplitude affects the wave's peak but doesn't change the wave's length or position.

Period of Sine Functions

The period of a sine function is the distance required for the wave to complete one full cycle of its pattern. The period tells us how frequently the wave repeats. For a function described by \( y = a \sin(bx + c) + d \), the period is calculated as \( \frac{2\pi}{b} \).
This mathematic formula helps us understand the wave's timing. It dictates how stretched or compressed the wave appears.
For the basic sine function \( y(x) = \sin x \), \(b = 1\), so the period is \( \frac{2\pi}{1} = 2\pi \). This means the wave repeats every \(2\pi\) units on the x-axis.

• A smaller period means more frequent oscillations.
• A larger period results in wider cycles.

Knowing the period is crucial for predicting wave behavior over time.

Average Value of Functions

In the context of trigonometric functions, the average value gives us the central tendency of the wave over one cycle. For sine functions in the format \( y = a \sin(bx + c) + d \), the average value per period is represented by \(d\).
This means that the horizontal line \(y = d\) will cut the wave in a manner that balances above and below the oscillations.
For the standard sine function \( y(x) = \sin x \), there is no vertical shift, so \(d = 0\). Hence, the average value is \(0\).
The average value can provide insights into the baseline level around which the wave oscillates.

• If \(d\) is positive, the whole wave shifts upwards.
• If \(d\) is negative, it shifts downwards.

Horizontal Shift of Functions

The horizontal shift in functions, often referred to as "phase shift," is the movement of a wave along the x-axis. In a trigonometric function of the form \( y = a \sin(bx + c) + d \), the horizontal shift is calculated as \( \frac{-c}{b} \).
This shift helps identify where the wave cycle begins along the x-axis; it influences the starting point of each wave cycle in its domain.
For \( y(x) = \sin x \), there is no horizontal phase shift because \(c = 0\). Thus, the computation \( \frac{-c}{b} = \frac{-0}{1} \) results in 0, meaning the wave begins at the usual starting point with no movement left or right.
Understanding horizontal shifts makes it easier to see how the waveform corresponds to changes in time or space.

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

Knowing this can help with modeling real-life periodic scenarios such as tides or sound waves.