Problem 4 - Precalculus Mathematics for Calculus

Question

Graph the function. $$f(x)=3+\sin x$$

Step-by-Step Solution

Answer

Graph the sine wave shifted up by 3 units, oscillating between 2 and 4.

Step 1: Understanding the Function

The function given is \( f(x) = 3 + \sin x \).This means it is a vertical shift of the basic sine function, \(\sin x\), upwards by 3 units. The sine function normally oscillates between -1 and 1 along the Y-axis, but with this function, it oscillates between 2 and 4.

Step 2: Identify Key Characteristics

The amplitude of the function is the same as the amplitude of \(\sin x\), which is 1. However, the midline is now \(y = 3\) because of the vertical shift. The period of the sine function is still \(2\pi\).

Step 3: Plotting the Sine Wave

Start by plotting the midline at \(y = 3\). Then, mark the function's maximum and minimum values at 4 and 2, respectively. Begin by plotting key points: where \(x = 0\), \(f(x) = 3\); \(x = \frac{\pi}{2}\), \(f(x) = 4\); \(x = \pi\), \(f(x) = 3\); \(x = \frac{3\pi}{2}\), \(f(x) = 2\); and \(x = 2\pi\), \(f(x) = 3\).

Step 4: Completing the Graph

Connect the key points with a smooth, wavy line to represent one full cycle, ensuring the curve passes through each plotted point smoothly. Extend the wave left and right as needed to sketch additional cycles, continuing the pattern established.

Key Concepts

Vertical Shift

A vertical shift in a function moves the graph of the function up or down along the Y-axis. In the function \(f(x) = 3 + \sin x\), the graph has been shifted vertically upwards by 3 units. This means wherever the sine function \(\sin x\) would normally be on the graph, you have to move every point up 3 units.
Instead of oscillating between -1 and 1 on the Y-axis, this shifted sine wave oscillates between 2 and 4.

• The highest point, which usually would be at 1, is now at 4.
• The lowest point, which usually would be at -1, is now at 2.

The line that represents the 'average' location between these oscillations is called the midline, and it is found at \(y = 3\) due to the vertical shift. This is a simple yet powerful transformation that helps visualize changes in trigonometric functions.

Amplitude

The amplitude of a trigonometric function reveals how 'tall' the waves are from the midline to either the peak or the trough. For a function like \(\sin x\), the amplitude is 1, meaning it extends 1 unit above and 1 unit below the midline.
In the shifted function \(f(x) = 3 + \sin x\), the amplitude remains the same even though the vertical shift changes the midline.

• Amplitude indicates the vertical stretch or shrink: Here, it is unchanged.
• A higher amplitude would stretch the wave vertically further from the midline, and a lower one would compress it closer.

Knowing the amplitude is key to understanding how the graph 'behaves' vertically. Despite any vertical shifts, the amplitude continues to show how far the function moves above and below the new midline.

Period of Sine Function

The period of a trigonometric function signifies the horizontal distance required for the function to complete one full cycle before it starts repeating.
For the basic sine function \(\sin x\), this period is \(2\pi\). This indicates that after \(2\pi\) units on the X-axis, the function's shape starts repeating itself.
In the function \(f(x) = 3 + \sin x\), the period remains \(2\pi\) despite the vertical shift.
The sine wave retains its width, so to speak, even after being shelved up by a vertical shift.

• The frequency, which is the reciprocal of the period, also remains unchanged.
• The vertical shift does not affect how "often" the sine waves come through; it affects only their position along the Y-axis.

Understanding the period is vital to predicting how a trigonometric graph repeats itself, and this remains consistent across vertical shifts.