Problem 64 - Algebra 2

Question

Graph each function in the interval from 0 to 2\(\pi\). \(y=\sin (x-\pi)+4\)

Step-by-Step Solution

Answer

The graph of the function \(y=\sin (x-\pi)+4\) in the interval from 0 to \(2\pi\) is a sine wave that is shifted \(\pi\) units to the right (a horizontal shift) and 4 units up (a vertical shift). The function oscillates between 3 and 5.

Step 1: Identify the basic function and transformations

The basic function is the sine function \(y = \sin(x)\). The function given in the exercise contains two transformations: a horizontal shift by \(\pi\) to the right (indicated by \(x-\pi\) in the function) and a vertical shift 4 units up (indicated by the +4 in the function).

Step 2: Plot the basic sine function in the given interval

The sine function \(y = \sin(x)\) oscillates between -1 and 1. In the interval from 0 to \(2\pi\), it starts from 0, goes up to 1 at \(\pi/2\), comes back to 0 at \(\pi\), goes down to -1 at \(3\pi/2\), and goes back up to 0 at \(2\pi\). Plot this function as a reference.

Step 3: Apply the horizontal shift

The horizontal shift moves the graph to the right by \(\pi\). This means that where the sine function was 0, it now becomes 0 at \(\pi\). Similarly, other points shift to the right by \(\pi\). Plot this on the graph.

Step 4: Apply the vertical shift

The vertical shift moves the graph up by 4 units. This means all the y-values will be increased by 4. For the values that were 0, they will now become 4. The values that were 1 and -1 will now become 5 and 3, respectively. Plot this on the graph.

Key Concepts

Sine Function

The sine function, represented mathematically as \(y = \sin(x)\), is one of the basic functions used in trigonometry. It is a periodic function, meaning it repeats its values in regular intervals or periods. The period of the basic sine function is \(2\pi\), which means it completes a cycle over this interval.

• The sine function oscillates between -1 and 1.
• It starts at 0, reaches its maximum value of 1 at \(\pi/2\), returns to 0 at \(\pi\), reaches its minimum value of -1 at \(3\pi/2\), and cycles back to 0 at \(2\pi\).

This wave-like pattern is what makes the sine function so useful for modeling periodic phenomena such as sound waves and tides.

Transformations

Transformations in mathematics refer to various operations that are applied to a function to produce a new function. These can shift, stretch, compress, or reflect the original graph. For the sine function, common transformations include reflection, shifts, and changes in amplitude or period.

• A basic transformation often involves a shift along the x-axis or y-axis, as well as potentially altering the graph's amplitude or frequency.
• Such transformations can help in adjusting the function to fit specific criteria or data points.

Understanding transformations is crucial in graphing because they allow us to visualize the effects of altering a function's parameters.

Horizontal Shift

A horizontal shift occurs when all points of a graph are moved along the x-axis by a certain distance. In the sine function, a horizontal shift is represented by a shift within the angle of the sine argument.For example, in \(y = \sin(x - \pi)\), the graph of \(y = \sin(x)\) is shifted horizontally to the right by \(\pi\) units.

• Prior to the shift, the basic sine wave starts at 0. After the horizontal shift, it starts at \(\pi\).
• Similarly, every other critical point on the graph (maximum, minimum, zeros) also shifts by this amount.

This adjustment changes the timing of the sine wave's peaks and troughs, which may be needed to align the function with external data or requirements.

Vertical Shift

A vertical shift involves moving the entire graph of a function up or down along the y-axis. In the context of the sine function, a vertical shift is a straightforward adjustment of the outputs' values.When we add a constant to the sine function, such as in \(y = \sin(x) + 4\), each point on the graph is moved vertically upward by that constant, 4 units in this case. This results in:

• The baseline of the sine wave, which originally oscillates between -1 and 1, now moves to a range between 3 and 5.
• All the zeros of the sine function move from 0 to 4.

Vertical shifts are often used to adjust the position of a wave-like graph to a desired range within a coordinate system, making them valuable for graph customization and application.