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Magazine Chapter 7: Problem 15
Graph the function for one period. Specify the amplitude, period, \(x\) -intercepts, and interval(s) on which the function is increasing. $$y=-2-2 \cos 3 \pi x$$
Short Answer
Expert verified
Amplitude: 2, Period: \(\frac{2}{3}\), x-intercepts: \(x = \frac{1}{3}\), Increasing on (0, \(\frac{1}{3}\)).
Step by step solution
01
Identify the Amplitude
The amplitude of a cosine function in the form \(y = a + b \cos cx\) is given by the absolute value of \(b\). Here, \(b = -2\), so the amplitude is \(|-2| = 2\).
02
Determine the Period
The period of a cosine function \(y = a + b \cos cx\) is calculated as \(\frac{2\pi}{c}\). In this function, \(c = 3\pi\), so the period is \(\frac{2\pi}{3\pi} = \frac{2}{3}\).
03
Calculate the x-intercepts
To find the \(x\)-intercepts, set \(y = 0\). Solve \(-2 - 2 \cos 3\pi x = 0\). This simplifies to \(\cos 3\pi x = -1\). The general solution for \(\cos \theta = -1\) is \(\theta = (2n+1)\pi\), where \(n\) is an integer. Substitute \(3\pi x\) for \(\theta\) to get \(3\pi x = (2n+1)\pi\), giving \(x = \frac{2n+1}{3}\). Within one period \(\left(0, \frac{2}{3}\right)\), the intercept is at \(x = \frac{1}{3}\).
04
Define Intervals of Increase
The function increases where the derivative is positive. Start by finding the derivative: \(y' = 6\pi \sin 3\pi x\). Solve \(6\pi \sin 3\pi x > 0\) to find the intervals of increase. Since \(\sin 3\pi x > 0\) when \(3\pi x\) is in \(\left(0, \pi\right)\), the interval for \(x\) is \(\left(0, \frac{1}{3}\right)\) within one period.
05
Graph the Function
Incorporate identified features in the graph: centerline at \(y = -2\), amplitude \(2\), period \(\frac{2}{3}\), and \(x\)-intercept at \(\frac{1}{3}\). Draw the curve reflecting these aspects from \(x=0\) to \(x=\frac{2}{3}\). Ensure the curve decreases from \(x=0\) to \(x=\frac{1}{3}\), then increases until \(x=\frac{2}{3}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Amplitude
The amplitude of a trigonometric function is essentially its height from the centerline to the peak or trough. It tells us how much the curve oscillates above and below its midline. For a cosine function in the form \( y = a + b \cos cx \), the amplitude is determined by the absolute value of the coefficient \( b \).
In the given function \( y = -2 - 2 \cos 3\pi x \), the amplitude is \( |-2| = 2 \). This means the maximum distance the graph will rise above and dip below its midline, which is at \( y = -2 \), is 2 units.
In the given function \( y = -2 - 2 \cos 3\pi x \), the amplitude is \( |-2| = 2 \). This means the maximum distance the graph will rise above and dip below its midline, which is at \( y = -2 \), is 2 units.
- The curve will reach a peak value of \( -2 + 2 = 0 \).
- The trough will go down to \( -2 - 2 = -4 \).
Period
The period of a trigonometric function is the length of one complete cycle before the pattern repeats itself. For the cosine function defined as \( y = a + b \cos cx \), the period is calculated using the formula \( \frac{2\pi}{c} \).
For the function \( y = -2 - 2 \cos 3\pi x \), since \( c = 3\pi \), the period becomes \( \frac{2\pi}{3\pi} = \frac{2}{3} \).
For the function \( y = -2 - 2 \cos 3\pi x \), since \( c = 3\pi \), the period becomes \( \frac{2\pi}{3\pi} = \frac{2}{3} \).
- This means every \( \frac{2}{3} \) unit along the x-axis, the cosine wave repeats its shape.
- The period dictates how many times the wave occurs within a specified interval, such as a single unit along the x-axis.
Cosine Function
The cosine function, denoted as \( \cos x \), is one of the fundamental trigonometric functions. It is periodic and oscillates between 1 and -1.
In its basic form, the function \( y = \cos x \) starts at maximum value 1 when \( x = 0 \), decreases through 0, reaches a minimum of -1, then returns through 0 back to 1 to complete a cycle.
For a transformed cosine function like \( y = -2 - 2 \cos 3\pi x \), several changes occur:
In its basic form, the function \( y = \cos x \) starts at maximum value 1 when \( x = 0 \), decreases through 0, reaches a minimum of -1, then returns through 0 back to 1 to complete a cycle.
For a transformed cosine function like \( y = -2 - 2 \cos 3\pi x \), several changes occur:
- The amplitude of 2 affects the height from the midline.
- The multiplication by \( -1 \) flips the graph vertically.
- The shift by -2 moves the entire graph downward.
- The factor of \( 3\pi \) compresses the period to \( \frac{2}{3} \).
Graphing
Graphing trigonometric functions requires careful attention to the characteristics of amplitude, period, and transformations like vertical shifts. For the function \( y = -2 - 2 \cos 3\pi x \), you need to plot the main attributes.
Start by identifying these key elements:
Start by identifying these key elements:
- A centerline at \( y = -2 \) due to the vertical shift.
- An amplitude of 2, marking the peak at \( y = 0 \) and trough at \( y = -4 \).
- A period of \( \frac{2}{3} \), indicating how often the cycle repeats.
- An x-intercept at \( x = \frac{1}{3} \) within a single period.
