Problem 5 - Calculus Concepts

Question

For each of the functions, state the amplitude, period, average value, and horizontal shift. \(\quad p(x)=\sin (2.2 x+0.4)+0.7\)

Step-by-Step Solution

Answer

Amplitude: 1, Period: \(\approx 2.857\), Average Value: 0.7, Horizontal Shift: \(\approx -0.182\)

Step 1: Understanding the Function

The given function is of the form \( p(x) = a \sin(bx + c) + d \), where \( a \) represents the amplitude, \( b \) affects the period, \( c \) influences the horizontal shift, and \( d \) shifts the graph vertically. Here, \( a = 1 \), \( b = 2.2 \), \( c = 0.4 \), and \( d = 0.7 \).

Step 2: Finding the Amplitude

The amplitude of a sine function is the absolute value of the coefficient \( a \). Thus, the amplitude of \( p(x) = \sin(2.2x + 0.4) + 0.7 \) is \( |1| = 1 \).

Step 3: Calculating the Period

The period of a sine function \( \sin(bx) \) is given by \( \frac{2\pi}{b} \). For the function \( p(x) = \sin(2.2x + 0.4) + 0.7 \), the period is \( \frac{2\pi}{2.2} = \frac{2\pi}{2.2} \approx 2.857 \).

Step 4: Determining the Average Value

The average value of the sine function is affected by the vertical shift \( d \). Therefore, the average value of \( p(x) = \sin(2.2x + 0.4) + 0.7 \) is \( 0 + 0.7 = 0.7 \).

Step 5: Finding the Horizontal Shift

The horizontal shift of the function \( a \sin(bx + c) + d \) is found by solving \( bx + c = 0 \). This gives \( x = -\frac{c}{b} \). Therefore, the horizontal shift of the function \( p(x) = \sin(2.2x + 0.4) + 0.7 \) is \(-\frac{0.4}{2.2} \approx -0.182 \).

Key Concepts

Amplitude

The amplitude of a trigonometric function like the sine function tells you how much it stretches or shrinks vertically. You can imagine it as the height of the wave. For any function of the form \( a \sin(bx + c) + d \), the amplitude is the absolute value of \( a \). This is because the amplitude is always a positive number that measures the distance from the highest point of the wave to the horizontal axis, or baseline.

• In our example, \( p(x) = \sin(2.2x + 0.4) + 0.7 \), the coefficient \( a \) is 1.
• Therefore, the amplitude is \( |1| = 1 \).

This means that the sine wave will rise and fall 1 unit above and below its midline, which is determined by the vertical shift given by \( d \). Understanding the amplitude is crucial for graphing sine waves as it affects the overall "height" of each cycle.

Sine Function

The sine function, represented by \( \sin(x) \), is a mathematical function that describes a smooth, repetitive oscillation. This function is a fundamental component of trigonometry. Being periodic in nature, the sine function is commonly used to model cyclical phenomena such as sound waves and tides.

• The basic sine function, \( \sin(x) \), oscillates between -1 and 1.
• It's crucial to recognize that the sine curve is symmetrical and links to the angle within a circle.

In the context of the function \( p(x) = \sin(2.2x + 0.4) + 0.7 \), modifications to the basic sine function occur, making it an ideal case study:

• The coefficient \( b = 2.2 \) affects the frequency of the cycles.
• It maintains the oscillatory patterns of sine but at a faster rate due to this coefficient.

Grasping the sine function's behavior is key to predicting how it shifts and stretches on a graph.

Periodicity

Periodicity refers to how often a sine wave repeats itself. It provides insight into the frequency of the cycles of the function. For a sine function \( \sin(bx) \), the period is calculated using \( \frac{2\pi}{b} \).

• In our function \( p(x) = \sin(2.2x + 0.4) + 0.7 \), \( b = 2.2 \).
• Thus, the period is \( \frac{2\pi}{2.2} \approx 2.857 \).

This calculation tells us that each complete wave cycle is approximately 2.857 units in length along the x-axis. The period determines the horizontal length of one oscillation and helps you to understand how frequently or rarely the sine value completes a full cycle. Recognizing periodicity helps with plotting the curve precisely and predicting its shape over different intervals.

Horizontal Shift

A horizontal shift moves the entire sine wave either left or right along the x-axis. This is determined by the value \( c \) in the function \( a \sin(bx + c) + d \). To find out precisely how much the function shifts, we solve for \( x \) when \( bx + c = 0 \).

• For \( p(x) = \sin(2.2x + 0.4) + 0.7 \), we substitute the values to get \( 2.2x + 0.4 = 0 \).
• This leads to a horizontal shift of \( x = -\frac{0.4}{2.2} \approx -0.182 \).

The function moves to the left by approximately 0.182 units. Understanding these shifts is essential when analyzing real-world data expressed through sine functions, as it indicates the phase or starting point of the wave. Completing these steps provides essential insights into how waves transform and behave under various conditions.