Problem 18 - Algebra 2

Question

Sketch one cycle of each sine curve. Assume \(a>0 .\) Write an equation for each graph. amplitude \(4,\) period 4\(\pi\)

Step-by-Step Solution

Answer

The equation of the sine curve with amplitude 4 and period \(4\pi\) is \(y = 4\sin(0.5x)\)

Step 1: Identify 'a' and 'b'

In this case, 'a' which represents the amplitude is given as 4. The period 'T' is given as \(4\pi\), we calculate 'b' using the formula \(b = 2\pi / T\). Substituting \(T = 4\pi\) we get \(b = 2\pi / 4\pi = 0.5\).

Step 2: Write the Equation of the Sine Curve

Now that we have the values for 'a' and 'b', we can write the equation of the sine curve. The equation thus becomes \(y = a \sin(bx) = 4\sin(0.5x)\)

Key Concepts

Understanding Amplitude

The amplitude of a sine curve refers to the peak value or maximum deviation from the central equilibrium position, which is typically the x-axis in a graph. In simpler terms, amplitude measures how "tall" or "short" the wave appears.
It can be thought of as the height from the centerline of the wave to the peak (or trough).

• In mathematical terms, amplitude is represented by the coefficient 'a' in the sine function equation: \( y = a \sin(bx) \).
• It is always a positive value that indicates the maximum vertical distance of the peak (or trough) from the centerline of the graph.

For our sine curve, the amplitude is given as 4, meaning the curve will reach 4 units above and below the centerline.
This helps visualize the oscillation range of the sine curve.

Determining the Period

A period is the distance over which the sine curve repeats itself. After this interval, the wave pattern repeats identically. To visualize this, imagine the wave as a cycle, starting at the origin and completing one full oscillation back to the starting point.
The period 'T' is crucial because it tells us how "stretched" or "compressed" the wave looks along the x-axis.

• In the equation \( y = a \sin(bx) \), the period is calculated using the formula \( T = \frac{2\pi}{b} \).
• For our example, 'b' was calculated to be 0.5, given by \( b = \frac{2\pi}{T} \), with \( T = 4\pi \).
• This implies that the sine curve completes one full cycle every \( 4\pi \) units along the x-axis, resulting in a wider spacing between repeated patterns.

This understanding helps you sketch your sine wave accurately by knowing the span over which it repeats.

Basics of Trigonometric Functions

Trigonometric functions like sine are fundamental to understanding wave patterns and oscillatory behaviors in mathematics.
These functions have wide applications, from analyzing sound waves to understanding electrical currents.

• The basic form of the sine function is \( y = \sin(x) \), which showcases a repeating pattern over intervals of \( 2\pi \).
• By manipulating the amplitude ('a') and period ('b'), as shown in \( y = a \sin(bx) \), the sine wave can be adjusted to fit specific requirements.
• The function oscillates between -1 and 1, but when scaled by 'a', its range becomes \([-a, a]\).

Understanding these adjustments makes it easier to apply sine functions to real-world scenarios, especially where periodic changes are observed.