Problem 67 - 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. $$ y=-2 \sin (4 x+\pi) $$

Step-by-Step Solution

Answer

The amplitude is 2, period is \( \frac{\pi}{2} \) and the phase shift is \( -\frac{\pi}{4} \). The function would need to be graphed over the interval \( -\frac{\pi}{4} \) to \( \frac{3\pi}{4} \) in the x-axis and between -2 and 2 in the y-axis for two complete periods.

Step 1: Identify A, B, C Values

The equation is of the form \(y=Asin(B(x+C))\). Here, A is the amplitude which is -2 (the negative makes the function 'upside down'). B affects the period of the function and is 4. C affects the horizontal shift of the function and is \(\pi\).

Step 2: Compute the Period of the Function

The standard period of a sine function is \(2\pi\). However, the period (P) of the given function can be altered by coefficient B according to the formula: \(P=\frac{2\pi}{B}\). So, substituting for B=4 in the formula we have: \(P = \frac{2\pi}{4} = \frac{\pi}{2}\)

Step 3: Determine the Phase Shift

The phase shift (D) is given by \(D = -\frac{C}{B}\). So, substituting for \(C=\pi\) and \(B=4\), we have: \(D = -\frac{\pi}{4}\)

Step 4: Select the Viewing Window

Considering that the amplitude is 2(-2 as a result of vertical flip) and one period as \(P = \frac{\pi}{2}\), we must show two periods for the function. So, choose the x-values from \(D = -\frac{\pi}{4}\) to \(D + 2P = -\frac{\pi}{4} + \pi = \frac{3\pi}{4}\) (this covers two periods). The y-values can be from -2 to 2, thus taking into consideration the amplitude. Use these values to graph the function over these intervals.

Key Concepts

Amplitude and Period of Sine Function

Understanding the amplitude and period of a sine function is fundamental in graphing these periodic waves. The amplitude of a sine function, represented mathematically as 'A' in the function format \(y = Asin(B(x+C))\), dictates the vertical stretch of the wave. In simple terms, the amplitude is the distance from the midline of the wave to its peak or trough. For example, an amplitude of 2 means the wave will reach 2 units above and below the center line, creating taller waves than the standard sine function, which has an amplitude of 1.

The period, influenced by the 'B' value, determines how long it takes for the function to complete one full cycle. The standard sine function has a period of \(2\text{π}\), repeating every \(2\text{π}\) radians. To find the period of a modified function, you use the formula \(P = \frac{2\text{π}}{B}\). If 'B' is greater than 1, the function will complete its cycles more quickly, resulting in more waves in the same length of the x-axis; if 'B' is less than 1, the cycles elongate. Hence, precision in determining these values ensures an accurate representation of the sine function on a graph.

Phase Shift

Phase shift in trigonometry refers to the horizontal displacement of a wave along the x-axis. When we modify a sine function to include a phase shift, we are essentially sliding the graph to the left or to the right. This is achieved by adding or subtracting a constant 'C' within the argument of the sine function (\(y = Asin(B(x+C))\)).

To calculate the phase shift, the formula used is \(D = -\frac{C}{B}\). This tells us how far and in which direction to shift the graph. If the phase shift is positive, the sine curve moves to the right; if negative, it moves to the left. This phase shift can change the starting point of a sine wave's cycle, which is crucial for aligning the wave with certain points on the x-axis, such as when modeling real-world phenomena like sound waves or tides.

Trigonometric Functions Graphing

Graphing trigonometric functions like sine, cosine, and tangent involves understanding their amplitude, period, phase shift, and vertical shift. Traditional graph paper or modern graphing utilities can both serve this purpose. When graphing, start by plotting the midline, which is the horizontal line equidistant from the peaks and troughs. The amplitude tells us how far to plot the peaks and troughs from this line, while the period indicates the distance between corresponding points, such as peak to peak or trough to trough, along the x-axis.

H4 Phase Shift and Viewing Window
Additionally, accounting for any phase shift is essential to determine where the sine curve should start. To effectively display the function, it’s important to choose an appropriate viewing window, encompassing at least one, but typically two full periods in the x-direction, and from the peak to the trough in the y-direction. Properly graphing these functions allows one to visualize and analyze the properties of trigonometric waves, enhancing understanding, especially in applications such as physics and engineering.

Precalculus Education

Precalculus serves as the foundation for the study of calculus, and it incorporates essential concepts like trigonometric functions graphing. Comprehending precalculus is vital for students aiming to succeed in higher-level mathematics and various science fields. A thorough grasp on the fundamental principles—including algebraic manipulation, functions and their properties, and the behavior of trigonometric functions—equips students with the analytical skills necessary for calculus.

Engaging with interactive tools, such as graphing calculators and educational software, can significantly aid in visualizing functions, while step-by-step learning helps solidify understanding. By breaking down complex ideas into manageable bits, educators can foster a deeper comprehension of precalculus concepts, allowing for a smoother transition into the world of calculus and beyond. The mastery of these basics is not only a stepping stone for further education but also lays the groundwork for numerous practical applications in science, technology, and engineering.