Problem 40 Sketch the graph of the function... [FREE SOLUTION]

Chapter 4: Problem 40

Sketch the graph of the function. (Include two full periods.) $$ y=\frac{1}{4} \sin x $$

Short Answer

Expert verified

The graph of the function $ y=\frac{1}{4} \sin x $ is a sine curve with amplitude 1/4 and period $ 2\pi $. It passes through the points $ (0, 0) $, $ (\frac{\pi}{2}, \frac{1}{4}) $, $ (\pi, 0) $, $ (\frac{3\pi}{2}, -\frac{1}{4}) $, and then repeats this pattern for a second period.

Step by step solution

Identify key properties of the function.

The function to be graphed is $y=\frac{1}{4} \sin x$, which is a sine function with amplitude of 1/4. The amplitude is the maximum height of the function, which is given by the coefficient in front of the sine function. The period of the function is $2\pi$, which is the standard period for a sine function.

Identify key points in one period.

The key points in one period of a sine function occur at intervals of $ \frac{\pi}{2} $ on the x-axis. These points divide the period into four equal parts: 0, $ \frac{\pi}{2} $, $ \pi $, $ \frac{3\pi}{2} $, and $ 2\pi $. The corresponding y-values for the sine function are 0, 1, 0, -1, and 0, respectively. For our function, we need to multiply these y-values by the amplitude, 1/4.

Sketch the graph.

Draw the x and y axis on a piece of graph paper. Plot the key points identified in step 2. Start with $ y=0 $ at $ x=0 $ and $ x=2\pi $, $ y=\frac{1}{4} $ at $ x=\frac{\pi}{2} $, $ y=\frac{-1}{4} $ at $ x=\frac{3\pi}{2} $, and y-values of 0 at $ x=\pi $ and $ x=2\pi $. Connect the points with a smooth, continuous curve. Repeat the pattern for a second period.

Repeat for a second period.

Extend the x-axis further to the right and repeat the pattern for a second period. That is, plot the points for $ x = 2\pi, \frac{5\pi}{2}, 3\pi, \frac{7\pi}{2}, $ and $ 4\pi $ using the same y-values as before. Connect these points with the same smooth, continuous curve.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Sine Function

The sine function, denoted as $y = \sin x$, is a fundamental trigonometric function that describes a smooth, periodic oscillation. It is known for its distinctive wave-like pattern, cycles every $2\pi$, and repeats this cycle indefinitely both left and right along the x-axis. In simple terms, if you imagine starting at the center of the wave, it rises to a peak, falls back through the center, dips to a trough, and returns once more to the center, completing a single cycle. This is how the basic sine function behaves, with its curve passing through key y-values of 0, 1, 0, -1, and 0 at equal intervals on the x-axis.

Exploring Amplitude of a Sine Function

The amplitude of a sine function determines how high and low the wave reaches. It is the vertical stretch or compression of the wave from its resting position or the horizontal axis. This is calculated by taking the absolute value of the coefficient in front of the sine function. For instance, in the function $y = \frac{1}{4} \sin x$, the amplitude is $\frac{1}{4}$. This means that the maximum height (or peak) of the function above the x-axis will be $\frac{1}{4}$, and the minimum height (or trough) below the x-axis will reach $-\frac{1}{4}$. Understanding amplitude is key to sketching accurate graphs of trigonometric functions.

Determining the Period of a Function

The period of a function is the length of one complete cycle of the waveform before it begins to repeat itself. For the standard sine function $y = \sin x$, the period is $2\pi$. This means it takes $2\pi$ units along the x-axis to complete one full oscillation. If the function includes a horizontal stretch or compression factor such as $y = \sin(bx)$, then the period changes and is calculated as $\frac{2\pi}{b}$. In our exercise, since there is no modification on the x-variable, the period remains $2\pi$, indicating that the wave returns to its starting point every $2\pi$ units on the x-axis.

Identifying Key Points on a Graph

Key points on a graph of a sine function help in accurately plotting the wave's shape. One complete cycle of the standard sine function can be divided into four key points based on the x-values: 0, $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, and $2\pi$. At these points, the y-values for a standard sine function breakdown to 0, 1, 0, -1, and 0 respectively, which shows the wave moving from center, to peak, back to center, downward to trough, and up to center again. For the function $y = \frac{1}{4} \sin x$,

At $x = 0$ and $x = 2\pi$, the y-value is 0.
At $x = \frac{\pi}{2}$, multiply the base y-value of 1 by $\frac{1}{4}$, resulting in a y-value of $\frac{1}{4}$.
At $x = \pi$, return to a y-value of 0.
At $x = \frac{3\pi}{2}$, the y-value becomes $-\frac{1}{4}$ as you multiply the base y-value of -1 by $\frac{1}{4}$.

Graphing these points will aid in sketching the wave over its cycle, and can then be repeated for accuracy over subsequent periods.

91影视

Sketch the graph of the function. (Include two full periods.) $$ y=\frac{1}{4} \sin x $$

Short Answer

Step by step solution

Identify key properties of the function.

Identify key points in one period.

Sketch the graph.

Repeat for a second period.

Key Concepts

Understanding the Sine Function

Exploring Amplitude of a Sine Function

Determining the Period of a Function

Identifying Key Points on a Graph

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