Problem 21
Question
Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=-\sin (\pi x / 2) $$
Step-by-Step Solution
Verified Answer
The graph is an inverted sine wave with a period of 4 and amplitude of 1, repeating every 4 units along the x-axis.
1Step 1: Understand the Parent Function
The parent function for the given equation is the sine function, which is generally denoted as \(y = \sin x\). The sine function has a period of \(2\pi\), an amplitude of 1, a range from -1 to 1, and it oscillates between these values.
2Step 2: Determine the Transformation
The given function is \(y = -\sin(\pi x / 2)\). This can be broken down into transformations of the parent sine function. The negative sign in front indicates a reflection across the x-axis. The factor \(\pi x / 2\) affects the period of the sine function, making it a factor of the horizontal stretching or compression.
3Step 3: Calculate the New Period
To find the new period, divide the original period of \(2\pi\) by the coefficient of \(x\) inside the sine function, which is \(\pi/2\). Thus, the new period is \(\frac{2\pi}{\pi/2} = 4\). This means the function completes a full cycle over the interval of 4 units on the x-axis.
4Step 4: Sketch the Key Points
Begin by identifying key points from one period of the sine function. Since the sign inverts the function: \( y = -\sin x \), the important points from \( y = -\sin x \) are: (0,0), \((2, -1)\), \((4,0)\) for one full period. These correspond to the start, midpoint, and end of one cycle.
5Step 5: Extend the Graph
To sketch more than one cycle, follow the periodic nature of sine. Repeat the pattern of the plotted points from Step 4 on either side along the x-axis, thereby extending to the left and right across multiple periods, respecting the wave's amplitude and period identified.
6Step 6: Answering for Major Features
The graph reflects a sine wave with an amplitude of 1, a period of 4, starts at the origin, and flips upside down because of the negative sign. The maximum and minimum values are 1 and -1 respectively, with zero crossings every 2 units.
Key Concepts
Sine FunctionPeriod of a FunctionGraph Transformations
Sine Function
The sine function is a fundamental trigonometric function, represented by the equation \( y = \sin x \). It graphs as a smooth, continuous wave that oscillates above and below the x-axis. This wave-like pattern is characterized by a few key features:
- Amplitude: The amplitude of the sine wave is the distance from the centerline (usually the x-axis) to the peak or trough of the wave. For \( y = \sin x \), the amplitude is 1. This means the graph reaches a maximum of 1 and a minimum of -1.
- Wavelength/Repetition: The wave repeats itself in a regular pattern, which we will discuss in more detail when considering the period of the function.
- Points of Zero Crossing: Where the sine wave crosses the x-axis determines the zero crossing points. In the parent sine function, this occurs at multiples of \( \pi \), like \( 0, \pi, 2\pi \)... and so forth.
Period of a Function
The concept of a period in trigonometry refers to the interval after which a function starts to repeat. For the sine function \( y = \sin x \), the period is \( 2\pi \). This means that every \( 2\pi \) units along the x-axis, the wave completes a full cycle and the pattern repeats.
When you transform the sine function, the period can change. For example, in the function \( y = -\sin(\pi x / 2) \), the period is altered. This change in period is found by dividing the original period \( 2\pi \) by the coefficient of \( x \), which in this case is \( \pi/2 \). Thus, the new period is \( \frac{2\pi}{\pi/2} = 4 \).
Understanding how to calculate and recognize changes in the period is crucial when graphing trigonometric functions and recognizing how their wave patterns shift on the x-axis.
When you transform the sine function, the period can change. For example, in the function \( y = -\sin(\pi x / 2) \), the period is altered. This change in period is found by dividing the original period \( 2\pi \) by the coefficient of \( x \), which in this case is \( \pi/2 \). Thus, the new period is \( \frac{2\pi}{\pi/2} = 4 \).
Understanding how to calculate and recognize changes in the period is crucial when graphing trigonometric functions and recognizing how their wave patterns shift on the x-axis.
Graph Transformations
Graph transformations such as reflections, translations, and stretches, are common operations applied to various functions, including trigonometric functions like the sine function. In the equation \( y = -\sin(\pi x / 2) \), multiple transformations occur.
- Reflection: The negative sign in front of \( \sin \) indicates a reflection over the x-axis. This means everything above the x-axis is flipped to below and vice versa, turning peaks into troughs and vice versa.
- Horizontal Stretch/Compression: The \( \pi x / 2 \) inside the sine function compresses the wave horizontally. Because of this, instead of the standard \( 2\pi \) cycle for a sine wave, the wave completes its cycle in 4 units.
Other exercises in this chapter
Problem 20
Suppose that \(f(x)=x^{4}, x \geq 0\). Find \(g(x)\) so that \(f \circ g=g \circ f\).
View solution Problem 20
Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line with slope \(-1\) and \(y\) -intercept \(
View solution Problem 21
Use a graphing calculator to graph \(f(x)=x^{2}, x \geq 0\), and \(g(x)=x^{4}, x \geq 0\), together. For which values of \(x\) is \(f(x)>g(x)\) and for which is
View solution Problem 21
Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line with slope \(1 / 2\) and \(y\) -intercept
View solution