Question

Graph each function using transformations or the method of key points. Be sure to label key points and show at least two cycles. Use the graph to determine the domain and the range of each function. $$ y=\sin (3 x) $$

Short answer

The domain is \((-\backsplash \infty, \backsplash \infty) \) and the range is \([-1,1]\).

Step-by-step solution

- Understand the Parent Function

The parent function of the given equation is \(y = \sin(x)\), which is a sine function. The basic sine function oscillates between -1 and 1, has a domain of all real numbers, and a period of \(2\pi\).

- Identify the Transformation

The function given is \(y = \sin(3x)\). Here, the coefficient 3 affects the period of the sine function. The period of \(y = \sin(kx)\) is \(\frac{2\pi}{k}\).

- Calculate the New Period

For the function \(y = \sin(3x)\), substitute \(k = 3\) into the period formula \(\frac{2\pi}{k}\). The period is \(\frac{2\pi}{3}\).

- Determine Key Points

To graph \(y = \sin(3x)\), determine key points for one period: the start (0,0), maximum (\(\frac{\pi}{6}, 1\)), intercept (\(\frac{\pi}{3}, 0\)), minimum (\(\frac{\pi}{2}, -1\)), and the end of the period (\(\frac{2\pi}{3}, 0\)).

- Extend to Two Cycles

Since the period is \(\frac{2\pi}{3}\), to show two cycles, the second cycle will end at \(\frac{4\pi}{3}\). Repeat the key points for the second cycle: start mid-cycle (\(\frac{2\pi}{3}, 0\)), maximum (\(\frac{5\pi}{6}, 1\)), intercept (\(\pi, 0\)), minimum (\(\frac{7\pi}{6}, -1\)), and end (\(\frac{4\pi}{3}, 0\)).

- Graph the Function

Plot the key points on a coordinate system and connect them smoothly to form the graph. Ensure to label the x-axis and y-axis with the key points.

- Determine Domain and Range

The domain of \(y = \sin(3x)\) is all real numbers \((-\backsplash \infty, \backsplash \infty) \). The range is the set of all y-values the function attains, which for the sine function is \([-1, 1]\).

Key concepts

Graphing Transformations

Graphing transformations are adjustments made to the parent function of a graph to reflect different changes, such as stretching or shifting. In our example, the parent function is the sine function, denoted as \(y = \sin(x)\). This function is transformed by multiplying the variable \(x\) by a constant, in this case, 3. This is a horizontal compression. To describe it more simply, this transformation will make the sine wave oscillate faster. The formula for the period of the transformed sine function \(y = \sin(kx)\) is \(\frac{2\pi}{k}\). Thus, for \(y = \sin(3x)\), the period changes to \(\frac{2\pi}{3}\). This means the sine wave will complete one cycle from 0 to \(\frac{2\pi}{3}\) instead of the usual \(0\) to \(2\pi\).

Key Points of Sine Function

Identifying key points of a sine function is crucial for graphing it accurately. Key points include the starting point, the maximum, the midpoints (intercepts), the minimum, and the ending point of one period. For the function \(y = \sin(x)\), the key points in one cycle are:

• Start: (0, 0)
• Maximum: \((\frac{\pi}{2}, 1)\)
• Midpoint (intercept): \((\pi, 0)\)
• Minimum: \((\frac{3\pi}{2}, -1)\)
• End: \((2\pi, 0)\)

When the function is transformed into \(y = \sin(3x)\), these points adjust to their new positions based on the period \(\frac{2\pi}{3}\). For one period, the updated key points become:

• Start: (0, 0)
• Maximum: \((\frac{\pi}{6}, 1)\)
• Midpoint (intercept): \((\frac{\pi}{3}, 0)\)
• Minimum: \((\frac{\pi}{2}, -1)\)
• End: \((\frac{2\pi}{3}, 0)\)

This cycle will repeat for subsequent periods, permitting accurate graphing over multiple cycles.

Domain and Range

Understanding the domain and range of a function helps define the set of possible x-values and y-values for the function. For the sine function, the domain is all real numbers because the sine function continues infinitely in both the positive and negative directions. Therefore, for \(y = \sin(3x)\), the domain remains all real numbers, written as (\(-\infty, \infty\)).
The range, however, considers the vertical extent of the function. The standard sine wave oscillates between \(-1\) and \(1\). This does not change no matter how the function is horizontally compressed or stretched. Therefore, the range for \(y = \sin(3x)\) remains \([-1, 1]\). Knowing these values ensures proper labeling and understanding of the sine wave's behavior on the graph.