Problem 50 - Algebra 2

Question

Sketch one cycle of each sine curve. Assume that \(a>0 .\) Then write an equation for each graph. amplitude \(4,\) period 1

Step-by-Step Solution

Answer

An equation representing the graph can be \(y=4\sin(2\pi x)\).

Step 1: Understand wave properties

The amplitude \(a\) is the peak value of the wave which in this case is 4. The period \(T\) is the interval over which the waveform shape repeats. In this case the period is given as 1. The full sine function can be given by the formula \(y = a \sin(bx + c)\), where \(a\) is the amplitude, \(b = \frac{2\pi}{T}\), is the frequency, and \(c\) is the phase shift.

Step 2: Draw the sine curve

Start by plotting both amplitude points at +4 (above x axis) and -4 (below x axis). Then plot a point at half the period, which is 0.5, at 0 (on x axis). Sketch the curve so that it goes from the peak at x=0, through the point in the center at x=0.5, and ends in the negative peak at x=1. You've now drawn one cycle of the sinusoidal wave.

Step 3: Write the equation

Using the sine function formula stated in step 1, as there is no phase shift, so c=0. Substitute amplitude \(a = 4\) and frequency \(b = \frac{2\pi}{1} = 2\pi\) into the equation to get \(y=4\sin(2\pi x)\).

Key Concepts

Sine Curve Sketch

To sketch a sine curve, we first need to understand the basic properties of a sine wave. The sine function produces a smooth, periodic oscillation. Its key properties include the amplitude and period. The amplitude is the height from the middle of the wave to its peak, while the period is the distance along the x-axis over which the wave completes one full cycle.

In our specific example, we are dealing with a sine wave with an amplitude of 4 and a period of 1. This means:

• The wave reaches as high as +4 and as low as -4 on the y-axis.
• The wave repeats every 1 unit along the x-axis.

When sketching the curve, starting with the peak at +4, then crossing the middle line (y=0) at half-period (x=0.5), and finally reaching the minimum at -4 at the period end (x=1) provides a full cycle of the wave. Remember that this pattern will keep repeating beyond one cycle.

Sine Function Equation

The equation for a sine function can be expressed in the form: \( y = a \sin(bx + c) \). This represents a standard sine wave modified by three parameters:

• Amplitude \(a\)
• Frequency component \(b\)
• Phase shift \(c\)

In the equation \( y = 4\sin(2\pi x) \), the amplitude \(a\) is 4, which indicates the height from the center line to the peak.
The frequency component \(b\) is \(2\pi\), derived from the relation \(b = \frac{2\pi}{T}\), where \(T\) is the period. Since \(T=1\), \(b=2\pi\). In this case, there’s no phase shift as \(c=0\). This equation models the periodic behavior we've sketched earlier.

Trigonometric Graphs

Trigonometric graphs show the oscillations of angles or cyclic phenomena. The sine function, a key trigonometric function, is especially useful for describing waves and periodic cycles. These graphs typically feature:

• Smooth, continuous curves that represent the wave-like properties of the function.
• The ability to model natural phenomena like sound waves and tides.
• Parameters such as amplitude, period, and phase shift which adjust the graph's shape.

The sine graph for our function \( y = 4\sin(2\pi x) \) illustrates a complete rise and fall every single unit along the x-axis. Depending on the need, sine graphs can be compressed or stretched along the x or y axes by altering their equations. Understanding how these transformations affect the sine curve is fundamental to mastering trigonometric functions.