Problem 58 - Contemporary Precalculus

Question

Graph the function. Does the function appear to be periodic? If so, what is the period? $$g(t)=|\sin t|$$

Step-by-Step Solution

Answer

If so, what is the period of the function? Yes, the function $$g(t) = |\sin t|$$ is periodic. The period of the function is $$\pi$$.

Step 1: Define the function

We are given the function $$g(t) = |\sin t|$$, which is the absolute value of the sine function.

Step 2: Sketch the graph of $$\sin t$$

Before we can graph $$|\sin t|$$, we need to recall the properties and graph of $$\sin t$$. The sine function is a periodic function with a period of $$2\pi$$, and it oscillates between $$-1$$ and $$1$$.

Step 3: Modify the graph of $$\sin t$$ to obtain the graph of $$|\sin t|$$

To graph $$|\sin t|$$, we take the graph of $$\sin t$$ and reflect all negative parts (values below the x-axis) upwards, turning them into positive values. This will result in a wave-like graph with peaks at $$1$$ and no negative values.

Step 4: Determine if the function appears to be periodic

Upon examining the graph of $$|\sin t|$$, we can see that it looks periodic as it repeats its pattern in a regular interval without any change in amplitude or frequency.

Step 5: If the function is periodic, find the period

We can observe that the graph of $$|\sin t|$$ completes one full cycle between $$0$$ and $$\pi$$. Therefore, the period of the function $$g(t) = |\sin t|$$ is $$\pi$$.

Key Concepts

Absolute Value Function

The absolute value function, denoted as \(|x|\), is a fundamental mathematical concept. It measures the distance of a number from zero on the number line, regardless of direction. This means it always returns a non-negative value. For example:

• The absolute value of 3 is 3, denoted \(|3| = 3\).
• The absolute value of -3 is also 3, denoted \(|-3| = 3\).

Applying this to functions, like the sine function, when we take the absolute value, all negative outputs are reflected above the x-axis into positive values. In the function \(g(t) = |\sin t|\), any part of the sine wave that dips below the x-axis is flipped upwards. Instead of oscillating from -1 to 1, the range changes to 0 to 1, while the form and properties significantly transform the graph's appearance.

Sine Function

The sine function, denoted as \(\sin t\), is crucial in trigonometry. It represents the y-coordinate of a point on the unit circle as a function of the angle from the positive x-axis. Here's a quick look at its core properties:

• It oscillates smoothly between -1 and 1.
• The sine of 0 is 0, \(\sin 0 = 0\).
• It peaks at 1 and troughs at -1.

This wave-like pattern makes it essential for modeling periodic processes. The complete cycle of the sine wave from 0 to \(2\pi\) reflects one full oscillation, making it inherently periodic. Understanding \(\sin t\) helps in grasping more complex wave functionalities, like those involving the absolute value of the sine, as seen in \(g(t) = |\sin t|\).

Trigonometric Graphs

When graphing trigonometric functions, like the sine and cosine, identifying key features is essential. Here's what to look out for:

• Amplitude: The maximum height from the center line, for the basic sine graph, it's 1.
• Period: The length of one complete cycle, which for \(\sin t\) is \(2\pi\).
• Frequency: How often the cycle repeats in a given interval.

In terms of transformation, \(g(t) = |\sin t|\) alters the sine graph by reflecting all negative values above the x-axis. This change brings about new insight into periodicity and symmetry, making the graph of \(|\sin t|\) wave-like and positive throughout. Learning these graphing skills aids in visualizing functions and understanding their behavior over intervals.

Function Periodicity

Periodicity is a vital concept in mathematics, especially concerning trigonometric functions. A function is periodic if it repeats its values at regular intervals or periods. Examining the periodicity involves:

• Checking Repetition: Does the function display a regular repeating pattern?
• Identifying the Period: The smallest interval over which the function repeats.
• Example: For \(\sin t\), the function repeats every \(2\pi\).

For the function \(g(t) = |\sin t|\), its periodicity changes because, unlike the \(2\pi\) period of \(\sin t\), \(g(t)\) completes one cycle over \(\pi\). This change arises from the nature of the absolute value, affecting the symmetry and duration needed for a full wave to repeat, offering students new perspectives on how functions can transform in continuous processes.