Question

Graph each function using transformations or the method of key points. Be sure to label key points and show at least two cycles. Use the graph to determine the domain and the range of each function. $$ y=3 \sin x $$

Short answer

Domain: \(( -\infty, \infty )\); Range: \([-3, 3]\)

Step-by-step solution

- Understand the parent function

The base function here is the sine function, which is given by \(y = \sin x\). The sine function has key characteristics: it has a period of \(2\pi\), crosses the x-axis at multiples of \(\pi\), reaches a maximum of 1 and a minimum of -1, and completes one cycle from \(0\) to \(2\pi\).

- Apply the transformation

The function given is \(y = 3 \sin x\), which is a vertical stretch of the parent function \(y = \sin x\) by a factor of 3. This means the amplitude of the sine function is multiplied by 3. Therefore, the maximum value will be 3 and the minimum value will be -3.

- Identify key points

For \(y = 3 \sin x\), the key points of one cycle (from \(0\) to \(2\pi\)) are: (0,0), (\(\frac{\pi}{2}\), 3), (\(\pi\), 0), (\(\frac{3\pi}{2}\), -3), and (\(2\pi\), 0). These points repeat every \(2\pi\) due to the periodic nature of the sine function.

- Plot the key points

Plot the points identified in step 3 on a coordinate plane. Mark the points (0,0), (\(\frac{\pi}{2}\), 3), (\(\pi\), 0), (\(\frac{3\pi}{2}\), -3), and (\(2\pi\), 0). Next, continue this pattern for another cycle by adding the period \(2\pi\) to each x-coordinate.

- Draw the graph

Connect the plotted points with a smooth, continuous wave to form the sine curve for two cycles. Ensure the wave reaches a maximum of 3 and a minimum of -3.

- Determine the domain and range

For the function \(y = 3 \sin x\), the domain is all real numbers \( ( -\infty, \infty ) \) because the sine function is defined for all \(x\)-values. The range is \([-3, 3]\) because the output values of the function will oscillate between -3 and 3 due to the vertical stretch.

Key concepts

Sine Function

The sine function is one of the fundamental functions in trigonometry. It is denoted as \( y = \sin x \). This function has a characteristic wave-like shape and is periodic, meaning it repeats its pattern over regular intervals. Key properties of the standard sine function include:

• A period of \( 2\pi \), indicating that the function repeats every \( 2\pi \) units.
• It crosses the x-axis at multiples of \( \pi \), such as \( 0, \pi, 2\pi \).
• The maximum value it reaches is 1, and the minimum value is -1.
• A complete cycle occurs from \(0\) to \( 2\pi \).

The parent function, \( y = \sin x \), serves as the foundation for understanding transformations like vertical stretch and changes in amplitude.

Vertical Stretch

A vertical stretch affects the amplitude of the sine function. The given function \( y = 3 \sin x \) represents the sine function vertically stretched by a factor of 3. Here’s what happens:

• Amplitude, which is the peak or maximum distance from the centerline (x-axis) to the peak of the wave, changes.
• Instead of peaking at 1 and -1, the stretched function peaks at 3 and -3.
• The sine curve still crosses the x-axis at the same points but extends higher and lower than the parent function.

One main advantage of understanding vertical stretch is simplifying graphing transformations by visualizing the extent functions are stretched along the y-axis.

Periodic Function

A periodic function repeats its values in regular intervals. The sine function, \( y = \sin x \), is periodic with a period of \( 2\pi \). This means the function's wave pattern repeats every \( 2\pi \) units along the x-axis. For the given function \( y = 3 \sin x \):

• The period remains \( 2\pi \), even though the amplitude has changed.
• Key points, such as peaks, troughs, and x-intercepts, repeat every \( 2\pi \).
• To graph more cycles, simply add the period \( 2\pi \) to the x-coordinates of the key points.

Understanding periodicity is crucial for predicting the behavior of trigonometric functions over long intervals.

Amplitude

Amplitude refers to the maximum absolute value of a periodic function. For the sine function, it is the distance from the midline (x-axis) to the peak of the wave. In the parent sine function, \( y = \sin x \), the amplitude is 1. However, in the transformed function \( y = 3 \sin x \):

• The amplitude is now 3.
• This change means the graph will reach a maximum value of 3 and a minimum value of -3.
• Amplitude dictates the range of the function, which in this case is \( [-3, 3] \).

Recognizing how the amplitude changes with transformations helps in accurately graphing and interpreting the modified functions.