Problem 27 - Algebra 2

Question

Graph each function in the interval from 0 to 2\(\pi\) $$ y=\frac{1}{2} \sin 2 x-1 $$

Step-by-Step Solution

Answer

The graph of the function \(y=\frac{1}{2} \sin 2x -1\) from 0 to \(2\pi\) is a sinusoidal wave with an amplitude of \(\frac{1}{2}\), period of \(\pi\), vertical shift of -1, and no horizontal shift.

Step 1: Identify Amplitude and Vertical Shift

The amplitude of the function is the absolute value of the coefficient of the sine function, which is \(\frac{1}{2}\), while the vertical shift is the constant term -1. Therefore, the graph oscillates between -1.5 and -0.5.

Step 2: Calculate the Period

The period of the function is given by \(\(2\pi\)/|B|\), where B is the coefficient of x. Here, B=2, therefore the period is \(\frac{2\pi}{2} = \pi\).

Step 3: Sketch Preliminary Points

Start graphing by plotting the midline of -1. Then, divide the period into four equal parts: \(0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi\). At each of these points calculate their corresponding y-values using the function...

Step 4: Complete the graph

After marking and connecting these points smoothly, continue the pattern for the rest of the interval up to \(2\pi\), and do not forget the function repeats every \(\pi\) units horizontally.

Key Concepts

Amplitude

In trigonometric functions like the sine function, the amplitude is a measure of how much the graph waves up and down from its centerline, or midline, which remains steady over the entire graph. For the equation in the exercise, the amplitude is the absolute value of the coefficient in front of the sine function. In this case, we have \(-1\/2\). This means the graph of the function will wave up to \(-0.5\) and down to \(-1.5\) from its midline, which is positioned at \-1\.

Understanding amplitude helps define the range of the graph, which is the span between its maximum and minimum values. The amplitude can tell you how "tall" the waves of the graph are relative to the midline.

Period

The period of a trigonometric function refers to how long it takes for the function to complete one full cycle before starting to repeat. In mathematical terms, it's the horizontal length of one complete wave from start to finish.

For the sine function \( y = \frac{1}{2} \sin 2x - 1 \), you determine the period by using the coefficient of \ x \ in the argument of the sine function. Mathematically, the period \( T \) is given by the formula \( T = \frac{2\pi}{|B|} \). Here, \( B = 2 \), so the period is \( \frac{2\pi}{2} = \pi \). This means the function completes its wave cycle every \( \pi \) units along the x-axis.

Knowing the period is crucial for correctly sketching the graph, as it tells you where to start and end one wave cycle before repeating it.

Vertical Shift

A vertical shift in a graph refers to moving the entire graph up or down along the y-axis. In the given function, \( y = \frac{1}{2} \sin 2x - 1 \), the vertical shift is determined by the constant term at the end, which is \-1\.

This indicates that the entire sine wave, including its midpoint and maximum/minimum amplitudes, is moved downward by 1 unit from the origin. The midline, which is normally on the x-axis for a standard sine function, has been lowered to \-1\ instead.

• The amplitude oscillates around the new midline located at \ y = -1 \.

The vertical shift helps to adjust the position of the wave on a graph without altering its shape or amplitude.

Interval

When graphing functions, especially trigonometric ones, specifying the interval is important to understand what portion of the graph you will observe. The interval essentially tells us which section of the x-axis we should be focusing on or measuring. In this exercise, you are asked to graph the function within an interval from 0 to \ 2\pi \.

This means you will only display the function for values of \ x \ starting from 0 and ending at \( 2\pi \). Within this interval, for the function given, because the period is \ \pi \ as calculated, you should expect to see two complete cycles of the sine wave.

The chosen interval affects the appearance of the graph, how many cycles of the function appear, and is crucial for understanding how to replicate the section of the graph you must draw.

Sine Function

The sine function is one of the most well-known trigonometric functions and forms the basis for understanding more complex trigonometric graphs. It is usually defined as \( y = \ sin x \) for the simplest form, which demonstrates a wave-like, periodical behavior over the interval of \( 0 \) to \( 2\pi \) by completing one full cycle.

For the given exercise \( y = \frac{1}{2} \sin 2x - 1 \), there's a combination of transformations applied to the basic sine function:

• The coefficient \( \frac{1}{2} \) alters the amplitude.
• The coefficient \ 2\ in front of \ x \ changes the period.
• The \-1\ constant shifts the graph vertically.

These transformations modify the sine curve's appearance, allowing us to stretch, compress, or relocate it across the coordinate system. Understanding these adjustments is key to mastering graphing trigonometric functions.