Problem 62 Sketch the graph of \(y=-3 \cos ... [FREE SOLUTION]

Chapter 4: Problem 62

Sketch the graph of \(y=-3 \cos \frac{3 x}{4},-2 \pi \leq x \leq 4 \pi\).

Short Answer

Expert verified

The graph of \(y = -3 \cos \frac{3x}{4}\) from \(-2\pi\) to \(4\pi\) is a wave that starts at \(-3\) at \(x = -2\pi\) and \(x = 4\pi\), with midpoint at \(y = 0\), and reaches the greatest height at \(y = 3\), at the mid of the period, and this pattern repeats approximately 2 times within this interval due to the period \(\frac{8\pi}{3}\).

Step by step solution

Identify the amplitude

The amplitude of the function, denoted by \(A\), is the magnitude of the maximum displacement from the equilibrium (midline in the graph). It's |a| in \(y = a \cos(bx)\). Here, \(a = -3\), so \(A = |-3| = 3\). The negative sign indicates a reflection about the x-axis.

Determine the period

The period, denoted by \(T\), of a cosine function is given by \(T = \frac{2\pi}{|b|}\) where \(b\) is the coefficient of \(x\) in \(y = a \cos(bx)\). Here, \(b = \frac{3}{4}\), so \(T = \frac{2\pi}{|\frac{3}{4}|} = \frac{8\pi}{3}\). This is the horizontal length for one complete cycle of the cosine function

Plot the amplitude and period

Plot the cosine function starting at \(x = -2\pi\), and ending at \(x = 4\pi\), with an amplitude of 3, and period \(\frac{8\pi}{3}\). Since we have a negative amplitude, the function starts at \(y = -3\) at \(x = -2\pi\) and \(x = 4\pi\), has midpoint at \(y = 0\), and reaches the maximum at \(y = 3\) in the mid of the period.

Sketch the complete graph within the given interval

Draw smooth, continuous, curved lines to connect amplitude and period plotted points to sketch the complete graph within the interval \(-2\pi \leq x \leq 4\pi\)

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Key Concepts

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

Amplitude in Trigonometry

The amplitude of a trigonometric function refers to the height from the midline to the peak or trough of the wave. In simpler terms, it measures how far the function's values vary from the center position.

For a function in the form of \( y = a \cos(bx) \), the amplitude is determined by the absolute value of the coefficient of \( \cos \), which is \( |a| \).

This value signifies the vertical stretch or compression of the graph.
If \( a \) is positive, the graph maintains its standard orientation.
If \( a \) is negative, the graph reflects over the x-axis, flipping upside down.

In our exercise, \( a = -3 \), therefore, the amplitude is \( 3 \). The negative sign tells us there is a reflection of the graph about the x-axis, thus starting the wave in the opposite direction.

Period of a Function

The period of a trigonometric function is the horizontal length required for the function to complete one full cycle. For a cosine function, the formula to find the period is \( T = \frac{2\pi}{|b|} \), where \( b \) is the coefficient of \( x \) in the expression \( y = a \cos(bx) \).

This value determines the distance between repeating patterns on the graph.

A bigger \( |b| \) results in a shorter period, causing more cycles to occur over a given interval.
A smaller \( |b| \) extends the period, making the waves stretch out more.

In the exercise provided, \( b = \frac{3}{4} \), thus the period \( T \) calculates to \( \frac{8\pi}{3} \). Hence, the cosine wave completes one full cycle within this interval.

Cosine Function

The cosine function, denoted as \( \cos(x) \), is one of the fundamental trigonometric functions. It's characterized by its wave-like pattern which starts at its maximum value when there is no phase shift (horizontal offset).

Some notable features of the cosine function include:

It typically starts at the peak and descends to the midline, reaching the trough before climbing back to the peak.
It is even, meaning it exhibits symmetry about the y-axis: \( \cos(-x) = \cos(x) \).
Standard cosine functions oscillate between -1 and 1. However, the range can be altered with different amplitudes.

In the function \( y = -3 \cos\left(\frac{3x}{4}\right) \), the negative sign outside the cosine alters the starting position, leading with a downward initial direction due to the reflection.

Reflection About X-Axis

Reflection about the x-axis is a transformation that flips the graph of a function over the x-axis. When graphing, this reflection occurs if the function's coefficient is negative.

For a trigonometric function like \( y = a \cos(bx) \):

If \( a < 0 \), each point on the curve is mirrored so that if originally it held a positive y-value, after reflection it holds a negative y-value, and vice versa.
This transformation is essential as it changes the orientation of the wave without affecting its amplitude or period. It's not about stretching but simply "flipping" over the zero line.

In our particular function \( y = -3 \cos\left(\frac{3x}{4}\right) \), the reflection means that instead of the cosine wave starting at its maximum value, it begins at the lowest point at \( y = -3 \). The graph is then a mirror of the normal cosine curve through horizontal motion.

91影视

Sketch the graph of \(y=-3 \cos \frac{3 x}{4},-2 \pi \leq x \leq 4 \pi\).

Short Answer

Step by step solution

Identify the amplitude

Determine the period

Plot the amplitude and period

Sketch the complete graph within the given interval

Key Concepts

Amplitude in Trigonometry

Period of a Function

Cosine Function

Reflection About X-Axis

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