Problem 104 Use a graphing utility to graph ... [FREE SOLUTION]

Chapter 4: Problem 104

Use a graphing utility to graph two periods of the function. $$y=3 \sin (2 x-\pi)+5$$

Short Answer

Expert verified

The graph of the function $y=3 \sin (2x-\pi) +5$ starts with the sine function, rescales the amplitude to 3, shifts the function to the right by $\frac{\pi}{2}$ units, shortens the period to $\pi$, and then shifts upwards by 5 units. Plotting this yields the graph of two periods of the function.

Step by step solution

Understand the basic sine function

The basic sine function, $y = \sin(x)$, has an amplitude of 1 (the distance from the maximum peak to the mean value), a period of $2\pi$ (the distance from one peak to the next), and it crosses the x-axis (i.e., it is zero) at $0, \pi, 2\pi, ...$. We will use these properties to plot $y=3 \sin (2x-\pi) +5$.

Identify amplitude, period, and phase shift

First, identify the changes made to the sine function. We can see that 3 scales the amplitude, (2x-$\pi$) alters the period and the phase shift, and 5 shifts the graph vertically. The amplitude of the function is 3 instead of 1 (from 3 \sin). The period is not $2\pi$, it is $\pi$, due to the (2x). The phase (horizontal shift) is $\frac{\pi}{2}$ to the right because of the (-$\pi$). Finally, the phase shift is up by 5 units due to (+5).

Plot the transformed sine function

Now, plot the transformed function using the identified amplitude, period, and phase shift. Start by plotting the basic sine curve with amplitude of 3 and period of $\pi$. Then shift this graph to the right by $\frac{\pi}{2}$ units. Finally, shift the graph upwards by 5 units. Remember to plot at least two periods of the function to fulfil the requirements of the exercise.

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Key Concepts

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

Sine Function

The sine function is a fundamental trigonometric function that describes a smooth, periodic oscillation. It's often written as \[ y = \sin(x) \]In its basic form, this function creates a wave that starts at the origin, reaches its maximum at $ \frac{\pi}{2} $, returns to zero at $ \pi $, reaches its minimum at $ \frac{3\pi}{2} $, and completes a full cycle back to zero at $ 2\pi $\.Some key attributes of the sine function:

It oscillates between -1 and 1.
The sine function is periodic with a standard period of $ 2\pi $\.
It is an odd function, meaning $ \sin(-x) = -\sin(x) $\.

In the exercise, we modify the basic sine function to $ y = 3 \sin(2x - \pi) + 5 $\, applying changes to its amplitude, period, phase, and vertical shift.

Amplitude

Amplitude refers to the height of the wave. In a sine function, it is the maximum distance the wave reaches from its central axis (the midline of the wave). For the basic sine function, the amplitude is 1.To find the amplitude of the function $ y = 3 \sin(2x - \pi) + 5 $, you look at the coefficient in front of the sine, which in this case is 3. This means the wave will reach 3 units above and 3 units below its midline.Some important points to remember:

A higher amplitude means a taller wave.
The amplitude is always a positive number.
The amplitude does not affect the period of the sine wave.

In our function, the sine wave is stretched vertically by a factor of 3 compared to the basic sine wave.

Phase Shift

The phase shift is the horizontal displacement of the graph of the sine function from its original position. In the function $ y = 3 \sin(2x - \pi) + 5 $, this shift is determined by the expression $ (2x - \pi) $\.To determine the phase shift, solve for $ x $ when the expression inside the sine function is equal to zero:$ 2x - \pi = 0 $which simplifies to $ x = \frac{\pi}{2} $\.This calculation indicates a phase shift of $ \frac{\pi}{2} $ units to the right.Key points on phase shift:

If the number inside the parenthesis is positive, the shift is to the left, but here it is negative, hence to the right.
Phase shift can change where the wave starts its cycle.
Phase shift affects the entire wave pattern horizontally.

In our context, the sine wave starts its cycle at the point $ \frac{\pi}{2} $ instead of the origin.

Period

The period of a sine function is the distance over which the wave repeats itself. In the standard sine function $ y = \sin(x) $, the period is $ 2\pi $\.For the function $ y = 3 \sin(2x - \pi) + 5 $, we calculate the period by considering the coefficient of $ x $ in the sine function, which is 2. The formula to determine the new period is:\[ \text{New period} = \frac{\text{Standard period}}{\text{Frequency}} = \frac{2\pi}{2} = \pi \]Some quick points about the period:

The smaller the period, the more frequent the peaks and troughs appear, meaning the wave is compressed horizontally.
The period determines how "long" it takes for a complete cycle of the wave.
Changes to the period do not affect amplitude or vertical shifts.

In this problem, the period is halved, so the sine wave completes its cycle every $ \pi $\, making it twice as frequent as the standard wave.

91影视

Use a graphing utility to graph two periods of the function. $$y=3 \sin (2 x-\pi)+5$$

Short Answer

Step by step solution

Understand the basic sine function

Identify amplitude, period, and phase shift

Plot the transformed sine function

Key Concepts

Sine Function

Amplitude

Phase Shift

Period

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