Problem 12 - Precalculus : Building Concepts and Connections

Question

Use your knowledge of vertical translations to graph at least two cycles of the given functions. $$g(x)=\sin x+\frac{3}{2}$$

Step-by-Step Solution

Answer

The graph of the function \(g(x) = \sin x + \frac{3}{2}\) is a sine wave that completes one cycle every \(2\pi\) units, has a peak at \(y = \frac{5}{2}\), a trough at \(y = \frac{1}{2}\), and is shifted upwards by \(\frac{3}{2}\) units from the standard sine function.

Step 1: Understand the basic graph of sine function

The foundational step is to understand how the basic graph of the sine function looks like. The sine function \(y = \sin x\) has a wavelength of \(2\pi\), amplitude of 1, with peaks at \(y=1\) and troughs at \(y=-1\). It's also important to note that the sine function completes one cycle in the interval \(0\) to \(2\pi\).

Step 2: Apply the vertical translation

Now that we have the basic sine function in mind, let's apply the vertical translation to it. A vertical translation does not affect the wavelength; it merely shifts the graph up or down. In this case, \(g(x) = \sin x + \frac{3}{2}\) shifts the standard sine function upward by \(\frac{3}{2}\) units. This means that now the peaks occur at \(y=1 + \frac{3}{2} = \frac{5}{2}\) and the troughs at \(y = -1 + \frac{3}{2} = \frac{1}{2}\).

Step 3: Graph the function

Now we have all info needed to graph the function. Start by drawing a horizontal line at \(y = \frac{3}{2}\) to represent the vertical shift. Then, graph the sine wave such that it oscillates between the new peak and trough heights. Don't forget that the function completes one cycle every \(2\pi\) units, so make sure to accurately portray this in the graph. Repeat the graph for at least two cycles, as the question asks.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry, often represented as \(y = \sin x\). It creates a wave-like shape and is known for its cyclical nature.
This function repeats values in a consistent pattern over a specific interval, known as a cycle.
Here are some key characteristics of the sine function:

• Periodicity: The sine function completes one full cycle in an interval of \(2\pi\) radians or 360 degrees.
• Amplitude: It varies between its maximum value of 1 and minimum value of -1.
• Zero Points: The function crosses the x-axis at multiples of \(\pi\) (0, \(\pi\), \(2\pi\), etc.).
• Peaks and Troughs: The highest point (peak) is at \(y = 1\) and the lowest point (trough) is at \(y = -1\).

Understanding these features helps in visualizing how the sine function behaves and provides a baseline for applying transformations, such as vertical translations.

Graphing Trigonometric Functions

Graphing trigonometric functions like the sine function involves plotting the values of the function over an interval.
This activity helps visualize how these functions oscillate between their peaks and troughs.
To graph a basic sine function:

• Start by identifying key points in its cycle. For the sine function, this includes the peak, trough, and zero points.
• Plot these points over a range, usually from 0 to \(2\pi\) radians or 360 degrees, to complete one cycle.
• Connect these points with a smooth, continuous curve that mirrors the wave-like nature of the sine function.

For transformed or translated sine functions, the graphing process is similar, but adjustments need to be made.
We'll consider the effect of these transformations, particularly vertical translations, in more detail next.

Vertical Shifts in Trigonometry

Vertical shifts in trigonometry, such as those applied to the sine function, involve moving the entire graph up or down.
These shifts do not alter the shape or period of the graph, just its position along the y-axis.
For example, consider \(g(x) = \sin x + \frac{3}{2}\):

• This function represents a vertical shift of the basic sine wave upwards by \(\frac{3}{2}\) units.
• The original peaks at \(1\) become \(\frac{5}{2}\), and the troughs at \(-1\) become \(\frac{1}{2}\).
• Draw a horizontal line at \(y = \frac{3}{2}\) to visualize the axis of the transformed graph.

Vertical translations are straightforward to apply and don't affect the periodic nature or the wavelength of the trigonometric functions, facilitating an easier understanding of how these transformations shift the graph vertically.