Problem 22 - Algebra 2

Question

Sketch one cycle of the graph of each sine function. $$ y=2 \sin \theta $$

Step-by-Step Solution

Answer

One cycle of the function \(y = 2 \sin \theta\) starts from 0, ascends to a height of 2 at \(\pi/2\), returns to 0 at \(\pi\), descends to a depth of -2 at \(3\pi/2\), and returns to 0 at \(2\pi\). The graph repeats after every \(2\pi\) units.

Step 1: Identify the amplitude

The amplitude is given by the coefficient of the sine function. Here we can see it is 2, so the function will ascend to a height of 2 and descend to a depth of -2.

Step 2: Identify the period

Since there is no modification to \(\theta\), the period of the function is \(2\pi\). This means the function will complete one cycle in the interval from 0 to \(2\pi\).

Step 3: Sketch the graph

Draw the x-axis (representing \(\theta\)) and the y-axis (representing \(y = 2 \sin \theta\)). Then draw a mark at each point corresponding to \(\pi/2\), \(\pi\), \(3\pi/2\), and \(2\pi\) on the x-axis. Now, starting from the origin, the function ascends to 2 at \(\pi/2\), returns to 0 at \(\pi\), descends to -2 at \(3\pi/2\), and returns to 0 at \(2\pi\). Connect these points smoothly to sketch one cycle of the graph.

Key Concepts

Amplitude

The amplitude of a sine function is an important concept that helps us understand how tall the waves of the graph are. In mathematical terms, the amplitude is the absolute value of the coefficient in front of the sine function.

For example, in the function \[y = 2 \sin \theta\]the coefficient is 2. This tells us that the graph will reach its highest point at 2 and its lowest point at -2.

The amplitude effectively measures the distance from the midline of the wave (usually the x-axis in simple sine functions) to the peak or the trough.

• When the amplitude is larger than 1, the sine wave stretches vertically.
• An amplitude less than 1 compresses the wave.
• If the amplitude is negative, the wave flips upside down, but the absolute value still determines how far it stretches or compresses.

Understanding amplitude is crucial as it impacts how we perceive the highs and lows of the wave, and this forms the foundation for graphing trigonometric functions.

Period

The period of a sine function tells us how long it takes for the wave to complete a full cycle. This is crucial in understanding the repetitive nature of waves.

In the case of the function\[y = 2 \sin \theta\]there are no additional coefficients modifying \(\theta\), which means it aligns with the standard sine function, whose period is \(2\pi\). This indicates that the wave completes one full oscillation from 0 to \(2\pi\).

The period is found using the formula \[\text{Period} = \frac{2\pi}{|B|}\]where \(B\) is the coefficient of \(\theta\).

• If \(B\) is greater than 1, the wave completes more cycles within the original \(2\pi\) interval, making the wave appear more compressed.
• If \(B\) is between 0 and 1, the wave takes longer to complete a cycle, stretching it out over a larger interval.

Knowing the period is pivotal for sketching accurate graphs and understanding trigonometric cycles in different scenarios.

Trigonometric Graphs

Trigonometric graphs, like those of sine and cosine functions, show how these functions behave and repeat over intervals. These graphs are essential for visualizing and analyzing wave-like phenomena.

When sketching a sine graph, as seen in\[y = 2 \sin \theta\]several key points guide its structure:

• The wave starts at the origin \((0, 0)\), ascends to the highest point (amplitude) at \(\pi/2\), returns to the midline at \(\pi\), reaches the lowest point (negative amplitude) at \(3\pi/2\), and goes back to the midline at \(2\pi\).
• It's important to mark crucial points on the x-axis: \(0, \pi/2, \pi, 3\pi/2, 2\pi\).
• Connecting these points smoothly gives one complete cycle of the sine wave.

This structure reflects the sine function's periodicity and symmetry.

These graphs provide a visual representation, making it easier to understand changes in properties like amplitude and period. Exploring these elements can help students grasp the underlying patterns and applications of trigonometric functions in real-world contexts.