Problem 47 - Algebra 2

Question

Find the period and amplitude of each sine function. Then sketch each function from 0 to \(2 \pi\) $$ y=-2 \sin 2 \pi \theta $$

Step-by-Step Solution

Answer

The amplitude of the function \(y=-2\sin2\pi\theta\) is 2 and the period is 1. Sketch of this function will be an inverted sine graph vertically stretched by a factor of 2 and horizontally compressed by a factor of \(2\pi\).

Step 1: Identify the Amplitude

The amplitude of a sine function is given by the absolute value of the coefficient of the sine term. In this case, the coefficient is -2. So, the amplitude is \(|-2|\) = 2.

Step 2: Identify the Period

The period of a sine function is given by the formula \(T = \frac{2\pi}{|B|}\) where \(B\) is the coefficient of the variable inside the sine. In this case, \(B = 2\pi\) So, \(T = \frac{2\pi}{|2\pi|}\) = 1.

Step 3: Sketch the Function

Consider a basic pattern for sin function which starts at 0, rises to 1 at \(\pi/2\), returns to 0 at \(\pi\), falls to -1 at \(\frac{3\pi}{2}\), and returns to 0 at \(2\pi\). Repeat this pattern cycle for the complete graph. The amplitude is the peak value, so stretch the graph vertically by a scale factor of amplitude which is 2. The graph is inverted as the coefficient of sin is -2. The period is 1, which means the complete cycle of the pattern is completed at x=1, rather than at \(2\pi\). Therefore, compress the graph horizontally by a scale factor of \(2\pi\). Now, you have a graph of the function \(y=-2\sin2\pi\theta\).

Step 4: Verify the result

To verify the result, plug in some values to the function \(y=-2\sin2\pi\theta\) and ensure the corresponding \(y\) value matches with the sketch. For instance, for \(\theta=0\), \(y\) should be 0 according to the sketch and the actual y-value will be \(-2\sin2\pi*0\) which is also 0. Therefore, the sketch is correct.

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function that is often used to model oscillatory phenomena like sound waves or tides. It is represented as \( y = ext{sin}(x) \). The function typically oscillates between -1 and 1. Depending on the modifications to the function, such as coefficients and shifts, its behavior can change considerably. In this lesson, we work with a modified version, \( y = -2 ext{sin}(2\ heta) \). Here, the negative sign inverts the graph, while the coefficients and arguments determine how the graph is stretched horizontally or vertically.A sine wave begins from 0, rises, falls, and returns to 0 over a period. Knowing this basic shape is crucial for understanding how changes in parameters affect the overall function.

Amplitude

Amplitude refers to the peak value of the sine wave, essentially how "tall" it is. It is calculated as the absolute value of the coefficient in front of the sine function.For the function \( y = -2 ext{sin}(2\pi \theta) \), the amplitude is \(|-2| = 2\). This means the wave reaches 2 units above and below the central axis, not accounting for the sign.

• The amplitude gives us insight into how much a wave oscillates up and down from its baseline.
• In practical terms, it shows the maximum extent of the oscillation.

If you're sketching a sine function, the perspective or perception of amplitude helps in adjusting the vertical scaling factor for an accurate graph.

Period

The period of a sine function is the distance required for the wave to complete one full cycle. It dictates the length of one full oscillation before the pattern repeats itself.For a typical sine function \( y = ext{sin}(x) \), the period is \( 2\pi \). But with variations like in\( y = -2 ext{sin}(2\pi \theta) \), we calculate the period using the formula \( T = \frac{2\pi}{|B|} \), where \( B \) is the coefficient of \( \theta \).In this case:

• The coefficient \( B = 2\pi \)
• The period \( T = \frac{2\pi}{2\pi} = 1\)

This indicates the function completes its cycle in length 1, meaning it compresses the typical wave structure for tighter oscillations. Understanding period helps determine how wide each cycle should be on a graph.

Trigonometric Graph Sketching

Graph sketching for trigonometric functions can initially seem complex, but it breaks down into methodical steps.Start with understanding the basic shape of the sine wave: it begins at 0, ascends, descends, and returns to 0. Repeat this sequence normally takes place over \( 2\pi \). However, in \( y = -2 ext{sin}(2\pi \theta) \), the period shortens to 1, compressing the wave horizontally.

• Apply the amplitude of -2 to stretch the wave vertically. This increases the peaks to 2 and troughs to -2.
• The negative sign inverts the wave, which flips it around its axis.

When graphing:- Plot the crucial points corresponding to \( heta = 0, 0.25, 0.5, 0.75, 1 \).- Align these to \( y \) values calculated using \( y = -2 ext{sin}(2\pi \theta) \).Visualizing each transformation in stages ensures an accurate representation of the sine function as parameters vary.