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Chapter 6: Problem 12

Determine the amplitude, the period, and the phase shift of the function and, without a graphing calculator, sketch the graph of the function by hand. Then check the graph using a graphing calculator. $$y=\sin \left(-\frac{1}{4} x\right)$$

Short Answer

Expert verified
Amplitude: 1, Period: \8\pi\, Phase Shift: 0. Sketch: Plot points and connect smoothly for one period.

Step by step solution

01

- Identify the Amplitude

The amplitude of a sine function \(y = A \sin(Bx + C)\) is the absolute value of A. In this case, the function is \(y = \sin \left(-\frac{1}{4} x\right)\). Since there is no coefficient before the sine term, the amplitude \(A = 1\).
02

- Determine the Period

The period of a sine function is given by \(\frac{2\pi}{|B|}\). Here, \(B = -\frac{1}{4}\). Thus, the period \(\text{P} = \frac{2\pi}{|-\frac{1}{4}|} = 8\pi\).
03

- Calculate the Phase Shift

The phase shift of a sine function is given by \(\frac{-C}{B}\). In this case, \C = 0\ as there is no horizontal shift evident in the function. Thus, the phase shift is \(\frac{0}{-\frac{1}{4}} = 0 \).
04

- Sketch the Graph by Hand

To sketch the graph of \(y = \sin \left(-\frac{1}{4} x\right)\), follow these steps:1. Start by plotting the key points for a sine function over one period \([0, 8\pi]\).2. The sine function will have the points \((0, 0)\), \((2\pi, 1)\), \((4\pi, 0)\), \((6\pi, -1)\), and \((8\pi, 0)\).3. Since the coefficient of x is negative, reflect these points horizontally.4. Connect these points smoothly to form the sine wave.
05

- Verify with a Graphing Calculator

Use a graphing calculator to input the function \(y = \sin \left(-\frac{1}{4} x\right)\) and check if the graph matches the hand-sketched graph. The key points should align correctly and confirm the amplitude, period, and phase shift computations.

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Key Concepts

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

Amplitude Calculation
The amplitude of a sine function reveals the height of the peaks and the depth of the troughs from the middle of the wave. In general, for a sine function represented as \( y = A \sin(Bx + C) \), the amplitude is given by the absolute value of \( A \).

To find the amplitude of \( y = \sin \left(-\frac{1}{4} x\right) \), notice that there is no multiplier in front of the sine function. This means the coefficient \( A = 1 \). Consequently, the amplitude is:

  • \textbf{Amplitude} = 1
Period Determination
The period of a sine function determines how long it takes for the function to complete one cycle. The formula to find the period is \( \frac{2\pi}{|B|} \), where \( B \) is the coefficient of \( x \) in \( y = A \sin(Bx + C) \).

For \( y = \sin \left(-\frac{1}{4} x\right) \), we identify \( B = -\frac{1}{4} \). Despite the negative sign, we take the absolute value:

The period \( P \) = \( \frac{2\pi}{|-\frac{1}{4}|} \) which simplifies to:

  • \textbf{Period} = 8\pi
Phase Shift Calculation
The phase shift tells us how much the function is shifted horizontally. It is obtained from the formula \( \frac{-C}{B} \) based on the general form \( y = A \sin(Bx + C) \). Here, \( C \) is the constant that indicates horizontal shift.

In the function \( y = \sin \left(-\frac{1}{4} x\right) \), there is no constant \( C \) term, so \( C = 0 \). Thus, the phase shift calculation simplifies to:

  • Phase shift = \frac{0}{ -\frac{1}{4} } = 0

This means there is no shift horizontally.
Hand Graphing Techniques
Graphing sine functions by hand involves plotting key points over one period and connecting them smoothly. Here's how:
  • Plot the key points for the sine function over one period \[0, 8\pi\]. For sine, these key points are:
    ( \(0, 0\), \(2\pi, 1\), \(4\pi, 0\), \(6\pi, -1\), \(8\pi, 0\)

  • Now, remember to reflect these points horizontally because of the negative coefficient of \( x \).
  • Connect these points smoothly to form a continuous wave.
  • Your sine wave will complete one cycle from \(0\) to \(8\pi\).
Graph Verification
After sketching the sine graph by hand, it is crucial to verify it using a graphing tool to ensure accuracy. Follow these steps:
  • Input the function \( y = \sin \left(-\frac{1}{4} x\right) \) into your graphing calculator.
  • Check the graph to see if the key points and shape match your hand-drawn graph.
  • Ensure that the graph conforms to the determined amplitude, period, and phase shift.
  • Confirm that the peaks, valleys, and zeros align precisely with your manual plot points.

This verification step ensures you have correctly interpreted and graphed the sine function.

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Most popular questions from this chapter

Make a hand-drawn graph of the function. Then check your work using a graphing calculator. $$f(x)=2^{-x}$$

Use a calculator in Exercises \(81-84,\) but do not use the trigonometric function keys. Given that $$\begin{array}{l}\sin 35^{\circ}=0.5736 \\\\\cos 35^{\circ}=0.8192 \\\\\tan 35^{\circ}=0.7002\end{array}$$ find the trigonometric function values for \(215^{\circ}\).

Classify the function as linear, quadratic, cubic, quartic, rational, exponential, logarithmic, or trigonometric. $$y=\sin x+\cos x[6.6]$$

Given that $$\begin{array}{ll}\sin 8^{\circ} \approx 0.1392, & \cos 8^{\circ} \approx 0.9903 \\\\\tan 8^{\circ} \approx 0.1405, & \cot 8^{\circ} \approx 7.1154 \\\\\sec 8^{\circ} \approx 1.0098, & \csc 8^{\circ} \approx 7.1853 \end{array}$$ find the six function values of \(82^{\circ} .\)

Find the function value. Round to four decimal places. $$\sin 118^{\circ} 42^{\prime}$$
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