Problem 34 - Precalculus Essentials

Question

Determine the amplitude of each function. Then graph the function and \(y=\cos x\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi\) $$y=-3 \cos x$$

Step-by-Step Solution

Answer

The amplitude of the function \(y = -3 cos x\) is 3. The graph of the function presents this amplitude as the maximum magnitude of the wave from its mean position. The graph also shows the comparison of this function with \(y = cos x\), showing differences in their amplitudes.

Step 1: Determine the Amplitude of the Function

The amplitude of a function is the absolute value of its greatest magnitude. For the function \(y = -3 cos x\), the amplitude is the absolute value of -3, which equals to 3.

Step 2: Graph the Given Function

In a graph, plot the coordinates for the function \(y = -3 cos x\) for the values of x between 0 and 2π. The points plotted will show how the function oscillates between -3 and 3 as x increases, reflecting the function's amplitude and period.

Step 3: Graph y = cos x

Next, on the same coordinate system, plot the coordinates for the function \(y = cos x\) for the values of x between 0 and 2π. It should oscillate between -1 and 1, showing the different amplitude of this function compared to the earlier one.

Step 4: Comparison of the Two Functions

Upon comparing the graphs of the two functions, \(y = -3 cos x\) and \(y = cos x\), it can be seen that the only difference between them is the amplitude. The function \(y = -3 cos x\) oscillates between -3 and 3, with a steeper gradient between the peaks and troughs, while \(y = cos x\) oscillates between -1 and 1.

Key Concepts

Graphing Trigonometric Functions

When we talk about graphing trigonometric functions, we're usually looking at how these functions behave visually on a coordinate plane. In the case of cosine and sine functions, this often resembles a wave-like pattern. For the function \(y = -3 \cos x\), it produces a wave that flips vertically compared to the standard cosine wave due to the negative sign. The coefficient before the cosine function affects the amplitude—the height of the wave's peaks and depth of its troughs. To graph \(y = -3 \cos x\), you must:

• Identify the amplitude, which is 3 in this case.
• Plot the cosine function oscillating between -3 and 3 on the y-axis for x values ranging from 0 to \(2\pi\).
• Observe how each complete cycle of the wave reaches its highest and lowest points.

Seeing how the function behaves on a graph makes it easier to understand periodic behavior—the repeating cycles of the function—which is intrinsic to trigonometric functions.

Period of Trigonometric Functions

The period of a trigonometric function like cosine is the length it takes to complete one full cycle of its wave. For \(y = \cos x\), the period is \(2\pi\). This means that every \(2\pi\) units, the function returns to its starting point and begins a new cycle. For any cosine function of the form \(y = a \cos(bx + c)\), you can find the period using the formula \(\frac{2\pi}{|b|}\).In the function \(y = -3\cos x\), the period remains \(2\pi\) as the coefficient of \(x\) (which is 1) does not change. This consistency in period allows us to predict the function's behavior over different intervals and can help with understanding and solving more complex trigonometric problems. In real-life applications, this periodicity appears in phenomena like sound waves and tides, where repeated cycles are a key characteristic.

Comparison of Cosine Functions

When comparing cosine functions, look primarily at alterations in amplitude and phase shifts (horizontal shifts) because these determine the graph's shape and placement. With \(y = -3\cos x\) and \(y = \cos x\), the primary difference is amplitude.

• \(y = \cos x\) has an amplitude of 1, as it oscillates between -1 and 1.
• \(y = -3 \cos x\) multiplies these values by -3, leading to an amplitude of 3, meaning it swings between -3 and 3.

The other critical factor is the vertical flip induced by the negative sign in \(-3\cos x\), providing a steeper oscillation compared to the gentler wave of \(\cos x\). Both functions have the same period of \(2\pi\) and are anchored at similar points over one complete cycle. By analyzing these components, one better understands how different parameters affect the waveform of trigonometric functions.