Problem 32 - Precalculus: A Concise Course

Question

Graph \(f\) and \(g\) on the same set of coordinate axes. (Include two full periods.) $$ \begin{array}{l} f(x)=\sin x \\ g(x)=\sin \frac{x}{3} \end{array} $$

Step-by-Step Solution

Answer

The graph of \(f(x)=\sin x\) completes two cycles between \(0\) and \(4\pi\), while the graph of \(g(x)=\sin \frac{x}{3}\) completes two cycles between \(0\) and \(12\pi\) due to its period being stretched by a factor of 3. This is shown graphically by elongation along the x-axis of the graph of \(g(x)=\sin \frac{x}{3}\) compared to the graph of \(f(x)=\sin x\)

Step 1: Review the properties of the sine function

The sine function, \( \sin x \), has a period of \(2\pi\) which means it completes one full cycle over an interval of \(2\pi\). The graph will start at the origin (0,0), then peak at \((\pi/2,1)\), comes back to 0 at \((\pi,0)\), then trough at \((3\pi/2,-1)\), and finally completes a cycle at \((2\pi,0)\). So, the graph of \(f(x)=\sin x\) should be drawn to show these characteristics over two periods.

Step 2: Understand how changes in the argument affects the sine function

The function \(g(x)=\sin \frac{x}{3} \) effectively stretches the period of the sine function by a factor of 3. This means the function will complete one cycle over an interval of \(6\pi\) instead of \(2\pi\). This will affect the positions of the peaks and troughs, so the graph of \(g(x)=\sin \frac{x}{3}\) should be drawn to reflect these changes.

Step 3: Graph the functions

Draw the graphs of \(f(x)=\sin x\) and \(g(x)=\sin \frac{x}{3} \) onto the same set of coordinate axes. Ensure to label the axes clearly and indicate the period of each function on the graph. For \(f(x)=\sin x\), the graph should complete two cycles between \(0\) and \(4\pi\). For \(g(x)=\sin \frac{x}{3}\), the graph should complete two cycles between \(0\) and \(12\pi\)

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry. It is a periodic function, meaning it repeats its values in a regular interval or period. The most basic form of the sine function is expressed as \( f(x) = \sin x \). This function takes an angle \( x \), measured in radians, and maps it to a point on the unit circle. The result is the vertical coordinate of that point, which ranges from -1 to 1.
One of the essential characteristics of the sine function is its smooth, wave-like graph, which starts at the origin \((0,0)\) in the coordinate plane. As \(x\) increases from 0 to \(2\pi\), the function passes through a series of critical points: it reaches 1 at \((\pi/2, 1)\), returns to 0 at \((\pi, 0)\), dips to -1 at \((3\pi/2, -1)\), and climbs back to 0 at \((2\pi, 0)\). This pattern repeats every \(2\pi\), forming the basis of its periodic nature.

• The sine wave's amplitude is the maximum distance from the x-axis, which is 1.
• The sine function is symmetric about the origin, making it an odd function.

Period of a Function

The period of a function is the interval over which the function completes one full cycle before it begins to repeat itself. For the sine function \( f(x) = \sin x \), the period is \(2\pi\). This means that every \(2\pi\) units along the x-axis, the pattern of the function's graph repeats itself.
However, when the argument of the sine function is modified, as in \( g(x) = \sin \frac{x}{3} \), the period changes. In this case, the function is stretched horizontally by a factor that is the reciprocal of the coefficient of \(x\) in the argument. The new period can be calculated as \( 2\pi \times b \), where \( b \) is the factor in the argument modification \( \frac{x}{3} \). This results in a period of \(6\pi\), meaning it takes \(6\pi\) units in the x-direction for the function to complete a full cycle.

• The period of the basic sine function is always \(2\pi\).
• Modifying the coefficient in the argument of \(x\) alters the period.
• The graph expands or contracts horizontally depending on the change in the period.

Graphing Trigonometric Functions

Graphing trigonometric functions like the sine function involves plotting its characteristic wave pattern across the coordinate plane. You start by identifying the function's key points, such as its intercepts, peaks, and troughs, over one period, and then repeat this pattern for each subsequent period.
For \( f(x) = \sin x \), you would graph this function across \(0\) to \(4\pi\) to cover two periods. The graph begins at the origin, peaks at \((\pi/2, 1)\), comes back to zero at \((\pi, 0)\), reaches a low at \((3\pi/2, -1)\), and then returns to zero to complete the cycle at \((2\pi, 0)\).
When graphing \( g(x) = \sin \frac{x}{3} \), the adjustments made to the argument change the function's period, expanding it to \(6\pi\). For two complete cycles, this graph must extend to \(12\pi\). Here, you'll notice that while the peaks and troughs occur less frequently along the x-axis compared to \( f(x) = \sin x \), the height of these peaks and troughs remains constant.

• Label axes clearly when graphing to avoid confusion.
• Indicate the period on the axis as a guide.
• Graphing helps visualize how changes in a function's equation affect its shape and period.