Problem 32 Graph one full period of each fu... [FREE SOLUTION]

Chapter 4: Problem 32

Graph one full period of each function. $$y=-4 \cos \left(\frac{3 x}{2}+2 \pi\right)$$

Short Answer

Expert verified

To graph the given function, begin by identifying the key features: The function has an amplitude of 4, so it will reach up to -4 and down to 4. Its period is $ \frac{4 \pi}{3} $, so one full cycle will occur within an interval of this size. Lastly, the phase shift is left by $ \frac{4 \pi}{3} $, so the graph should be shifted this amount to the left from the standard cosine graph.

Step by step solution

Identify the Amplitude

The coefficient in front of the cosine function is the amplitude. The amplitude is the height from the midline of the graph to the peak or to the trough. In this case, it is $ -4 $. Because it's negative, it means the cosine graph will be flipped about the horizontal axis. Therefore, the highest point will be at -4 and the lowest at 4.

Identify the Frequency

The number multiplying $ x $ inside the cosine function affects the function's frequency, which determines how many cycles the graph completes in a given interval. Here, the frequency is $ \frac{3}{2} $. The period is given by $ \frac{2 \pi}{\text{frequency}} = \frac{2 \pi}{\frac{3}{2}} = \frac{4 \pi}{3} $.

Identify the Phase Shift

A phase shift is caused by a change in the function's input which shifts the graph horizontally. It鈥檚 given by $ \frac{\text{shift inside cosine}}{\text{frequency}} $. Here it's $ \frac{2 \pi}{\frac{3}{2}} = \frac{4 \pi}{3} $. Because we have a positive sign in front of $ 2 \pi $, we shift to the left.

Draw the graph

You'll first draw the regular cosine graph. Then flip it over the midline because of the negative amplitude. After that, change the period length to $ \frac{4 \pi}{3} $ and shift it $ \frac{4 \pi}{3} $ units to the left because of the phase shift.

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Key Concepts

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

Understanding Amplitude in Trigonometric Functions

When graphing trigonometric functions like cosine, it's crucial to grasp what amplitude means. The amplitude is represented by the coefficient in front of the cosine or sine function. It measures the distance from the midline of the wave to its peak or trough.

For the function given, the amplitude is $-4$. This negative sign does something specific: it flips the graph over the horizontal axis. Normally, the highest point would be positive, but because of this negative amplitude, the highest point is actually at $-4$ and the lowest at $+4$.

Remember these key points about amplitude:

It affects the height of the graph.
A negative amplitude will flip the graph.
To find it, look at the coefficient in front of the trigonometric function.

Understanding amplitude helps you know how "tall" and in which direction your graph faces.

Exploring Frequency and Its Effect on Graphs

Frequency affects how many cycles a trigonometric graph completes in a specific interval. In our example, the frequency is determined by looking at the multiplier next to the variable inside the cosine function. Here, it's $\frac{3}{2}$.

Frequency heavily influences the period of the graph, which is the length of one complete cycle. The formula to find the period is $\frac{2\pi}{\text{frequency}} = \frac{2\pi}{\frac{3}{2}} = \frac{4\pi}{3}$. This tells us that the graph completes one full cycle every $\frac{4\pi}{3}$ units of the x-axis.

Key things to remember about frequency:

Higher frequency means more cycles in the same space.
It alters the period of the function.
You can control the "wavelength" of the graph through frequency.

Grasping frequency allows you to predict how the graph will behave in a given interval.

Phase Shift and Horizontal Movement

Phase shift refers to how a graph moves horizontally on the coordinate plane. It's affected by changes to the input value of the function. In our case, inside the cosine function, we have $+2\pi$.

The phase shift is calculated using the formula $\frac{\text{shift inside cosine}}{\text{frequency}} = \frac{2\pi}{\frac{3}{2}} = \frac{4\pi}{3}$. Here, the shift is positive, so we move the graph to the left by $\frac{4\pi}{3}$ units.

Crucial insights about phase shift:

A positive value shifts the graph left, while a negative value shifts it right.
It changes the starting point of the cycle on the x-axis.
Phase shifts can be tricky, but they ensure the graph starts exactly where it's supposed to.

Understanding phase shift is necessary to accurately determine where the wave begins on the graph.

91影视

Graph one full period of each function. $$y=-4 \cos \left(\frac{3 x}{2}+2 \pi\right)$$

Short Answer

Step by step solution

Identify the Amplitude

Identify the Frequency

Identify the Phase Shift

Draw the graph

Key Concepts

Understanding Amplitude in Trigonometric Functions

Exploring Frequency and Its Effect on Graphs

Phase Shift and Horizontal Movement

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