Problem 27 - A Graphical Approach to Precalculus with Limits

Question

Graph each function over the interval \([-2 \pi, 2 \pi] .\) Give the amplitude. $$y=-2 \sin x$$

Step-by-Step Solution

Answer

Amplitude is 2; graph from \(-2\pi\) to \(2\pi\), reflecting and stretching by 2.

Step 1: Identify Amplitude

The amplitude of a sine function is the absolute value of the coefficient in front of the sine term. In this function, the coefficient of \( \sin x \) is \(-2\). Therefore, the amplitude is \( | -2 | = 2 \).

Step 2: Determine the Interval

The specified interval over which you need to graph the function is \([-2\pi, 2\pi]\). This means that the graph should start from \(-2\pi\) and end at \(2\pi\) on the x-axis.

Step 3: Understand the Sine Wave Transformations

A sine function typically has a standard shape. Here, \(y = -2 \sin x\) combines a few transformations: the \(-\) sign indicates a reflection over the x-axis and the \(2\) indicates there is a vertical stretch of the graph by a factor of 2.

Step 4: Plot the Basic Sine Function

Start by plotting the points of the standard \(y = \, sin\, x\) for key angles within one period (from \(-2\pi\) to \(2\pi\)). The function completes its basic cycle between 0 and \(2\pi\), therefore, mark the main critical points: (0,0), (\(\pi/2\),1), (\(\pi\),0), (\(3\pi/2\),-1) and (\(2\pi\),0).

Step 5: Apply the Transformations

Reflect the basic sine wave across the x-axis and apply the vertical stretch by multiplying each of these y-values by 2. For example, the point (\(\pi/2\),1) will become (\(\pi/2\),-2) after reflection and stretching. Repeat for other critical points within the interval.

Step 6: Complete the Graph

Once you have the points for one cycle, continue the pattern to complete the graphing over the interval \([-2\pi, 2\pi]\). Since a sine graph is periodic, extend the points found in Step 5 backward to \(-2\pi\) using the same transformations.

Key Concepts

Understanding Amplitude in Sine Functions

Amplitude, in the context of sine functions, is a measure of how much the wave stretches or compresses vertically. It essentially tells us how "tall" or "short" the wave is. To find it, look at the coefficient in front of the sine term. In our function, \( y = -2 \sin x \), the coefficient is \(-2\). Though we ignore the negative sign when calculating amplitude, because amplitude is always positive. Thus, the amplitude here is:

• The absolute value of \(-2\), which is \(2\).

This means that the wave will reach 2 units above and below the horizontal axis. Imagine it like a musical note resonating in a broader scale but with the same fundamental sound.

Sine Wave Transformations

Transformations alter the basic shape of the sine wave to fit specific conditions. For the function \( y = -2 \sin x \), the transformations involve:

• Reflection: The negative sign before the \(\sin x\) indicates a reflection over the x-axis. This means, where the sine wave originally went upwards, it will now go downwards.


• Vertical Stretch: The coefficient \(2\) is applied to the absolute function. It means the wave is stretched vertically. This makes the peaks and valleys taller by a factor of 2.

These transformations help reshape the sine curve, displaying variations of the fundamental wave in diverse forms.

Graphing Intervals for Sine Functions

Graphing intervals set the boundaries within which the function is plotted. The interval ensures that we see complete cycles of behavior. For our function

• The interval is \([-2\pi, 2\pi]\).

This includes starting from the left at \(-2\pi\), moving through the origin at \(0\), and ending at \(2\pi\). This full stretch captures multiple cycles of the sine function, perfectly displaying both its periodic nature and transformations. As you traverse this range, you discern the repetitive peaks and troughs reflective of the function's pattern.

Exploring Vertical Stretch

Vertical stretch elongates the sine wave vertically, altering its appearance. By emphasizing the peaks and troughs, it modifies how high or low the wave travels. With our equation:

• The factor of \(2\) means every y-value is doubled from its original position if there was no vertical stretch.


• For example, a point initially at height \(1\) becomes \(2\), and a point at \(-1\) becomes \(-2\).

In the graph of \(y = -2 \sin x\), the vertical stretch notably transforms the standard sine wave into a more pronounced wave, clearly depicted in its graphical representation over the set interval.