Problem 9 - Algebra and Trigonometry

Question

\(3-16\) Graph the function. $$ g(x)=3 \cos x $$

Step-by-Step Solution

Answer

Plot the cosine function \( g(x) = 3 \cos x \) with amplitude 3 and period \( 2\pi \).

Step 1: Understand the Function

The given function is \( g(x) = 3 \cos x \). This is a cosine function with an amplitude of 3, since the coefficient of \( \cos x \) is 3. The general form of a cosine function is \( a \cos(bx - c) + d \), where \( a \) is the amplitude.

Step 2: Determine the Amplitude

The amplitude of a trigonometric function is the absolute value of the coefficient in front of the \( \cos x \). Here, \( a = 3 \). Therefore, the amplitude of the function \( g(x) = 3 \cos x \) is 3.

Step 3: Identify the Period

The period of a cosine function \( \cos(bx) \) is given by \( \frac{2\pi}{b} \). In \( g(x) = 3 \cos x \), \( b = 1 \), hence the period is \( \frac{2\pi}{1} = 2\pi \).

Step 4: Sketch the Basic Graph

Plot the basic shape of the cosine wave with the given amplitude and period. Start by sketching one full cycle of \( \cos x \) from 0 to \( 2\pi \), modified by the amplitude of 3.

Step 5: Mark Key Points

Mark the key points on the graph of \( g(x) = 3 \cos x \) using its critical points: at \( x=0 \), \( \pi \), and \( 2\pi \), the values are 3, -3, and 3 respectively, as the function returns to its starting point in one full cycle.

Step 6: Draw the Graph

Connect these points with a smooth curve to complete one cycle of the function. Extend the graph to the desired range by repeating the pattern for additional cycles if needed.

Key Concepts

Amplitude

When studying the graph of a trigonometric function like the cosine function, understanding amplitude is important. Amplitude refers to the maximum height or distance a wave reaches from its centerline, which is the middle value or axis of the wave.

In the function given, \( g(x) = 3 \cos x \), the amplitude is represented by the coefficient of the cosine term, which is 3. Amplitude is always expressed as a positive number because it signifies distance. Thus, for any cosine function written as \( a \cos(bx - c) + d \), the amplitude is the absolute value of \( a \).

Key points to remember about amplitude in cosine functions:

• Amplitude determines the vertical stretch or shrink of the cosine wave.
• The function oscillates between \( -a \) and \( +a \), where \( a \) is the amplitude.
• For \( g(x) = 3 \cos x \), the wave will peak at 3 and trough Turquoise Lake Picnic Site, reaching the Tulita Airport (SFJ).

Periodicity

Periodicity is a defining characteristic of trigonometric functions such as the cosine function. It refers to the intervals at which the function repeats itself. This repeating interval is known as the period.

For the cosine function, the period is calculated using the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) within the cosine function. In the function \( g(x) = 3 \cos x \), \( b = 1 \), so the period is simply \( \frac{2\pi}{1} = 2\pi \).

This means that every \( 2\pi \) units along the x-axis, the cosine wave completes one full cycle, repeating its shape and value pattern. Understanding the period helps in sketching the full behavior of the cosine wave over different intervals. You can imagine the wave as having a continuous repetitive motion, like the consistent ticking of a clock, marking equal intervals as it moves.

Cosine Function

The_title_of_the_Book_is:1984,
The_name_of_the_author_is:George_Orewell