Problem 71 - Precalculus: A Concise Course

Question

Use a graphing utility to graph the function. Include two full periods. Be sure to choose an appropriate viewing window. $$ y=-0.1 \sin \left(\frac{\pi x}{10}+\pi\right) $$

Step-by-Step Solution

Answer

The graph of the function \(y=-0.1 \sin \left(\frac{\pi x}{10}+\pi\right)\) displays two complete periods in the window running from \(x=-10\) to \(x=30\) and from \(y=-0.15\) to \(y= 0.15\).

Step 1: Identify the parameters

Identifying the parameters, you got: the amplitude \(A=-0.1\), the coefficient \(B = \frac{π}{10}\) and the phase shift \(C = π\).

Step 2: Calculate the period

The period is obtained by dividing \(2π\) by the absolute value of \(B\), which gives \(\frac{2π}{\frac{π}{10}} = 20\). This means that one cycle of the graph repeats every 20 units.

Step 3: Choose the viewing window

To display two full periods of the function, you should choose an interval for x that spans 40 length units, because the period is 20. Given that the function has a phase shift of π, the beginning of the first period is not at \(x=0\) but at \(x=-\frac{π}{\frac{π}{10}}=-10\). So, you could choose a window running from \(x=-10\) to \(x=30\). For y, considering that the amplitude is -0.1, the maximum and minimum values that y can reach are 0.1 and -0.1 respectively. Therefore, the window can run from -0.15 to 0.15.

Step 4: Sketch the function

Use a graphing utility to sketch the function. Remember to set the viewing window as determined. You can now see two full cycles of the function's graph.

Key Concepts

Period of a Sine Function

Understanding the period of a sine function is crucial when dealing with trigonometric graphs. The period defines how long it takes for the sine wave to complete one full cycle before it starts repeating itself. Mathematically, for a basic sine function defined as \( y = \text{sin}(x) \), the period is \( 2\text{π} \).

In the context of transformations, the function \( y = A \text{sin}(Bx + C) \) has a period that is determined by the value of B. Specifically, the period is calculated using the formula \( \text{period} = \frac{2\text{π}}{|B|} \), where |B| is the absolute value of B. For the given function \( y = -0.1 \text{sin}\big(\frac{\text{\text{π}} x}{10} + \text{π}\big) \), the period is \( 20 \) units. This means one complete wave of the sine function is drawn out over 20 units along the x-axis before it starts repeating.

Amplitude of a Sine Function

The amplitude of a sine function measures the wave's height from the central axis (horizontal axis) to its maximum or minimum. In a standard sine function like \( y = \text{sin}(x) \), the amplitude is 1 because the graph reaches 1 unit above and below the horizontal axis.

When looking at a function of the form \( y = A \text{sin}(Bx + C) \), the amplitude is represented by the absolute value of A. In our exercise, the amplitude of the function is the absolute value of -0.1, which is 0.1. This means the graph of this sine wave will peak at 0.1 units above the central axis and dip to 0.1 units below it, creating a relatively flat wave compared to the standard sine function.

Phase Shift in Trigonometry

A phase shift refers to the horizontal translation of a trigonometric function. In essence, it is how much the function is shifted left or right on the graph. For the sine function in the form \( y = A \text{sin}(Bx + C) \), the phase shift is calculated by solving for x in the equation \( Bx + C = 0 \), which gives \( x = -\frac{C}{B} \).

In our given exercise, with \( y = -0.1 \text{sin}\big(\frac{\text{\text{π}} x}{10} + \text{π}\big) \), we have a phase shift of \( -\frac{\text{\text{π}}}{\frac{\text{\text{π}}}{10}} = -10 \). This means the start of the sine wave is shifted 10 units to the left on the x-axis. Understanding phase shifts is important for accurately sketching graphs and interpreting their transformations.

Graphing Calculator Usage

A graphing calculator is a powerful tool for visualizing trigonometric functions and their transformations. To use a graphing calculator effectively, one must first input the function in the correct format. For our exercise, you'd enter \( y = -0.1 \text{sin}\big(\frac{\text{\text{π}} x}{10} + \text{π}\big) \).

Next, you set the viewing window. Based on the calculated period and amplitude, as well as the phase shift, adjust the window to encompass at least two full cycles of the wave and slightly beyond the maximum and minimum y-values. For the given function, a suitable x-range might be from -10 to 30, and a y-range of -0.15 to 0.15. Finally, use the graphing feature to produce a visual representation of the function. This visual aid can be incredibly helpful for comprehending the effects of amplitude, period, and phase shifts on the sine wave.