Problem 17 - Algebra 2

Question

Sketch one cycle of each sine curve. Assume \(a>0 .\) Write an equation for each graph. amplitude \(\frac{1}{3},\) period \(\pi\)

Step-by-Step Solution

Answer

The equation of the sine function with amplitude \(\frac{1}{3}\) and period \(\pi\) is \(y=\frac{1}{3} \sin(2x)\). The function starts at `x=0, y=0`, reaches its peak at \(x=\frac{\pi}{4}\), returns to 0 at `x=\pi/2`, reaches the lowest point at `x=3\pi/4`, and completes one cycle at `x=\pi`.

Step 1: Understanding amplitude and period of sine function

The standard sine function can be written as \(y=a \sin(bx)\) where `a` is the amplitude and `b` is a value which determines the period of the function. If `a` is greater than 1, it stretches the function vertically and if it is a fraction between 0 and 1, it compresses the function vertically. The value of `b` changes the period of function. The standard period of the sine function is \(2\pi\) . When `b` value is set to \(\frac{2\pi}{T}\), `T` being the desired period, it alters the period of the function.

Step 2: Form the equation

Here the amplitude `a` is given as \(\frac{1}{3}\) and period `T` is given as \(\pi\). Hence according to the sine function equation \(y=a \sin(bx)\), we replace `a` with \(\frac{1}{3}\) and `b` with \(\frac{2\pi}{T}\) i.e., \(\frac{2\pi}{\pi}=2\). Now our equation becomes \(y=\frac{1}{3} \sin(2x)\)

Step 3: Sketch the graph

To sketch the graph, start from `x=0` and `y=0` because sine of 0 is 0. The graph will reach its peak at \(\frac{1}{4}\) of the period, which is at \(x=\frac{\pi}{4}\), it returns to 0 at `x=\pi/2`, reaches the lowest point at `x=3\pi/4`, and completes one full cycle at `x=\pi`. The maximum and minimum values of y will be +1/3 and -1/3 respectively, as given by the amplitude.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry. It is a periodic function, which means it repeats its values in regular intervals. The standard sine function is denoted by \( y = \sin(x) \). This function produces a smooth, wave-like graph, which is why sine and other trigonometric functions are often called wave functions.

A key characteristic of the sine wave is its ability to model periodic phenomena, such as sound waves or tides. It starts from zero, rises to a maximum value, falls back to zero, drops to a minimum value, and returns to zero, completing one cycle. This distinctive pattern is crucial for understanding the behavior of waves in physics, engineering, and many applied sciences.

• Sine starts at 0
• Maximum is 1
• Minimum is -1
• Repeats every \(2\pi\)

Understanding the sine function forms the basis for exploring more complex trigonometric functions.

Amplitude

Amplitude refers to the height of the wave from its central axis, the horizontal line that passes through its middle. For the sine function \( y = a \sin(bx) \), the amplitude is represented by the absolute value of \( a \).

In terms of the graph, the amplitude dictates how "tall" or "short" the sine wave appears. A larger amplitude means a taller wave, and a smaller amplitude means a shorter wave. It's important to understand that amplitude only affects the vertical stretch of the graph, not its period or horizontal aspects.

• Amplitude is the absolute value of \( a \)
• Vertical stretch or compression
• Doesn't affect the period
• Determines peak and trough values

So, if the amplitude is \( \frac{1}{3} \), as in the given exercise, the graph oscillates between \( \frac{1}{3} \) and \( -\frac{1}{3} \). This results in a sine curve that looks compressed vertically compared to the standard sine curve, which oscillates between -1 and 1.

Period of a Function

The period of a sine function refers to the horizontal length over which the function completes one cycle. For the standard sine function \( y = \sin(x) \), this period is \(2\pi\).

However, when you modify the equation to \( y = a \sin(bx) \), the period is determined by the coefficient \( b \). To find the new period, use the formula \( \text{period} = \frac{2\pi}{b} \).

• Standard period: \(2\pi\)
• New period with \( b \): \( \frac{2\pi}{b} \)
• Affects the horizontal stretch or compression

In our example, \( b = 2 \), resulting in a period of \( \frac{2\pi}{2} = \pi \). This means the sine curve completes its cycle twice as fast compared to the standard sine wave, resulting in more cycles over the same horizontal span.

Graphing Sine Functions

Graphing a sine function requires understanding its amplitude and period, as they dictate the shape and length of the curve. Starting from the general function \( y = a \sin(bx) \), you can graph it by following these steps:

1. **Identify Parameters**: Note the values of \( a \) and \( b \). For example, \( a = \frac{1}{3} \) and \( b = 2 \) for our equation.

2. **Plot Key Points**: Start at \( x = 0 \). The peak points, zero crossings, and trough points depend on the period and amplitude.

3. **Draw the Curve**: Connect these points with a smooth, wave-like line to form the sine wave.

• Peak at \( \frac{\text{Period}}{4} \)
• Crosses zero midway
• Completes cycle by the period

In practice, for \( y = \frac{1}{3} \sin(2x) \):
- It starts from zero at \( x = 0 \)
- Peaks at \( x = \frac{\pi}{4} \) with a value of \( \frac{1}{3} \)
- Crosses back to zero at \( x = \frac{\pi}{2} \)
- Reaches a trough at \( x = \frac{3\pi}{4} \) with a value of \( -\frac{1}{3} \)
- Completes a full cycle back to zero at \( x = \pi \)
With this understanding, graphing becomes much more intuitive and straightforward.