Problem 2 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The function \( f(x) = 3 + \sin x \) is a sine wave shifted upwards by 3 units, oscillating between 2 and 4.

Step 1: Analyze the Function

The given function is \( f(x) = 3 + \sin x \). This function is a transformation of the basic sine function. The basic sine function, \( \sin x \), oscillates between -1 and 1. Adding 3 to the sine function shifts the entire graph of \( \sin x \) vertically upwards by 3 units.

Step 2: Identify Key Characteristics

Identify the maximum, minimum, and range of the function. Since \( \sin x \) oscillates between -1 and 1, the transformation 3 + \sin x shifts this range to 2 to 4. Therefore, the minimum value is 2 and the maximum value is 4. The midline, which is normally the x-axis (y = 0) for \( \sin x \), is shifted to y = 3.

Step 3: Graph the Vertical Shift

Start by drawing the midline of the function, which is y = 3. This is where the center of the wave will be located. Draw a horizontal line across the graph at y = 3 to represent this midline.

Step 4: Plot Key Points and Sine Waves

Identify key x-values such as 0, \( \pi/2 \), \( \pi \), \( 3\pi/2 \), and \( 2\pi \) and plot corresponding points. For these points, calculate the values: \( f(0) = 3 + \sin(0) = 3 \), \( f(\pi/2) = 3 + \sin(\pi/2) = 4 \), \( f(\pi) = 3 + \sin(\pi) = 3 \), \( f(3\pi/2) = 3 + \sin(3\pi/2) = 2 \), and \( f(2\pi) = 3 + \sin(2\pi) = 3 \). Plot these points.

Step 5: Sketch the Sine Curve

Using the plotted points, draw a smooth curve indicating the wave-like pattern of the sine function. This curve should oscillate above and below the midline drawn at y = 3, touching the midline at x = 0, \( \pi \), \( 2\pi \), and so on, reaching maxima and minima at \( \pi/2 \) and \( 3\pi/2 \), respectively.

Step 6: Verify the Pattern

Ensure the wave follows the correct oscillation, matching the sine function behavior. The peaks should touch at y = 4, and the troughs should touch at y = 2. The wave should be periodic, repeating every \( 2\pi \). Check that each cycle (from one maximum to the next) is consistent and matches the periodic nature of sine.

Key Concepts

Vertical Shifts in Trigonometry

In trigonometry, vertical shifts occur when a constant is added or subtracted from a trigonometric function. This results in the whole graph moving up or down the y-axis. Take the function \( f(x) = \, ext{a constant} \, + \, ext{a function} \). Here, the constant causes the graph to move vertically.

For instance, \( f(x) = 3 + \sin x \) moves the sine wave up by 3 units. Normally, the sine wave oscillates around the x-axis, but this shift moves it to oscillate around the new line \( y = 3 \). This is known as the midline. You can see this vertical shift clearly when you graph the function.

• Original sine function oscillates between -1 and 1.
• With a vertical shift, it oscillates between 2 and 4.

Recognizing vertical shifts is crucial to graphing trigonometric functions accurately.

Sine Function Transformations

The transformation of the sine function involves several steps, the first being identifying shifts, stretches, and reflections. The basic sine function, \( \sin x \), has a repeating pattern, typically described by its amplitude, period, phase shift, and vertical shift.

In our example, \( f(x) = 3 + \sin x \), the transformation simply includes a vertical shift. However, understanding other transformations is important. Consider that transformations can

• Change the amplitude by multiplying \( \sin x \) by a factor, altering the height of the wave.
• Change the period by multiplying \( x \) by a factor, reducing or extending the cycle duration.
• Include phase shifts by adding a constant inside the function, shifting the wave horizontally.

In our graph, the amplitude stays 1, the period remains unchanged at \( 2\pi \), and there is no horizontal phase shift.

Trigonometric Function Characteristics

Trigonometric functions have key characteristics: amplitude, period, frequency, phase shift, and vertical shift. These define their shape and oscillatory behavior.

Considering \( f(x) = 3 + \sin x \), its characteristics are:

• **Amplitude:** The natural wavelength amplitude of sine is 1. In this case, it remains 1, as there's no multiplication altering it.
• **Period:** The sine function has a regular cycle, completing every \( 2\pi \).
• **Vertical Shift:** We've moved the focus line up to \( y = 3 \) to center our wave around this line.
• **Maximum and Minimum Values:** This shift changes the highest and lowest y-values to 4 and 2, respectively.

Each of these characteristics helps in analyzing and graphing the function confidently. Understanding them ensures accurate plotting and interpretation of trigonometric functions.