Problem 14 - Algebra and Trigonometry

Question

Graph the function. $$ h(x)=|\sin x| $$

Step-by-Step Solution

Answer

Graph an absolute sine wave with peaks at 1 and troughs at 0, repeating every \(2\pi\).

Step 1: Understanding the Function

The given function is the absolute value of the sine function, denoted as \(|\sin x|\). This means that for every value of \(#x#\), we will output the absolute value of \(#\sin x#\). As a result, any negative values taken by the sine function will be turned into positive values.

Step 2: Identifying Key Characteristics

The function \(|\sin x|\) inherits several characteristics from \(\sin x\). It has a period of \(2\pi\), meaning it repeats its pattern every \(2\pi\) units. However, unlike \(\sin x\), the function only takes non-negative values, ranging from \(0\) to \(1\).

Step 3: Determining Critical Points

The critical points occur at \((0,0)\), \(\left(\frac{\pi}{2}, 1\right)\), \((\pi, 0)\), \((\frac{3\pi}{2}, 1)\), \(2\pi, 0)\), and this pattern repeats for every \(2\pi\). These points will guide the graph's peaks and troughs.

Step 4: Plotting the Function

Start plotting the function using the critical points: begin at \((0,0)\), rise smoothly to \(1\) at \(\frac{\pi}{2}\), fall back to \(0\) at \(\pi\), rise again to \(1\) at \(\frac{3\pi}{2}\), and back to \(0\) at \(2\pi\). Repeat this wave-like pattern for further cycles while ensuring the function stays non-negative.

Step 5: Drawing the Complete Graph

With the points and shape in mind, the graph of \(|\sin x|\) will appear as a series of arches, starting from zero, rising to a peak, and returning to zero, without dipping below the \(x\)-axis. Extend this pattern both to the left and right on the axis to show multiple periods.

Key Concepts

Absolute Value Function

The absolute value function transforms any input to its non-negative form. In mathematical terms, this is written as \(|a|\), where \(a\) represents any real number.

• If \(a\) is positive or zero, \(|a| = a\).
• If \(a\) is negative, \(|a| = -a\), which makes it positive.

Understanding absolute value is crucial in graphing functions like \(|\sin x\|\).For every point where the sine function dips below the x-axis, the absolute value function flips it upwards so it remains positive.This results in a graph that has only non-negative values along the y-axis.

Sine Function

The sine function, represented by \(\sin x\), is fundamental in trigonometry. Its graph forms a wave-like pattern:

• It starts at zero, rises to one, back through zero, down to negative one, and back to zero as \(x\) progresses from zero to \(2\pi\).
• This cycle repeats every \(2\pi\), which defines its period.

When considering the graph of \(|\sin x|\), these negative dips below the axis are ignored, giving a different visual representation that consists only of positive values and zeros.

Periodicity in Trigonometry

Periodicity is an important concept when dealing with trigonometric functions. It refers to the repeating nature of functions over specific intervals.For \(\sin x\), the period is \(2\pi\), meaning the function's values repeat every \(2\pi\) units.For \(\ |\sin x|\), although the negative parts of the cycle are flipped upwards, the period remains the same—\(2\pi\).This periodic behavior implies that once you understand one cycle, you can predict the behavior indefinitely along the x-axis.

Critical Points in Graphing

When graphing any function, identifying critical points—where the function hits maximums, minimums, or crosses an axis—is vital.For \(|\sin x|\), the critical points are:

• \( (0, 0) \)
• \( \left( \frac{\pi}{2}, 1 \right) \)
• \( (\pi, 0) \)
• \( \left( \frac{3\pi}{2}, 1 \right) \)
• \( (2\pi, 0) \)

These critical points help to shape the graph's wave-like appearance, indicating where the graph's peaks and troughs are located. Properly identifying and connecting these points ensures accurate and smooth graph representation.