Problem 25 - Algebra and Trigonometry

Question

Find the amplitude and period of the function, and sketch its graph. $$ y=1+\frac{1}{2} \cos \pi x $$

Step-by-Step Solution

Answer

Amplitude: \( \frac{1}{2} \), Period: 2.

Step 1: Identify the standard cosine function

The standard form for a cosine function is \( y = A \cos(Bx + C) + D \), where \( A \) is the amplitude, \( B \) affects the period, \( C \) is the phase shift, and \( D \) is the vertical shift. For the given equation \( y = 1 + \frac{1}{2} \cos(\pi x) \), the cosine component is \( \frac{1}{2} \cos(\pi x) \).

Step 2: Determine the amplitude

Amplitude is the absolute value of the coefficient of the cosine term in the function. Here, the cosine coefficient is \( \frac{1}{2} \), thus the amplitude is \( |\frac{1}{2}| = \frac{1}{2} \).

Step 3: Determine the period of the function

The period of a cosine function is calculated by \( \frac{2\pi}{B} \), where \( B \) is the coefficient of \( x \). In the equation, \( B = \pi \), so the period is \( \frac{2\pi}{\pi} = 2 \).

Step 4: Determine the vertical shift and graph considerations

The function \( y = 1 + \frac{1}{2}\cos(\pi x) \) indicates a vertical shift of +1 due to the constant term. When graphing, the midline will be at \( y = 1 \), and it oscillates with an amplitude of \( \frac{1}{2} \) above and below this line with a period of 2.

Key Concepts

Amplitude

Amplitude is a crucial feature in trigonometric functions such as the cosine and sine functions. It tells us how far the function's values stretch above and below the midline, which is the horizontal line that the wave oscillates around. In a cosine function like \[ y = A \cos(Bx + C) + D \]where \( A \) is the amplitude, this value is the absolute value of the coefficient in front of the cosine function.
- So if we consider the function \( y = 1 + \frac{1}{2} \cos(\pi x) \), the amplitude is \( |\frac{1}{2}| = \frac{1}{2} \).

• The amplitude is always a positive value because it is a distance.
• This tells us how much the curve extends above and below the middle line.
• The graph of this function is "half a unit" above and below the line \( y = 1 \).

Recognizing the amplitude helps visualize the height and depth of the waves generated by trigonometric functions, making it easier to sketch and understand the graph.

Period

The period of a trigonometric function is another central aspect that gives us information about the wave's behavior. The period indicates how often the wave pattern repeats along the x-axis.
For the standard cosine function \( y = A \cos(Bx + C) + D \), the period can be found using the formula:\[ \text{Period} = \frac{2\pi}{B} \]Here, \( B \) is the coefficient of \( x \) in the cosine function. It compresses or stretches the graph horizontally. For our function \( y = 1 + \frac{1}{2} \cos(\pi x) \):

• We have \( B = \pi \).
• The period is calculated as \( \frac{2\pi}{\pi} = 2 \).

This means the cosine wave will repeat its pattern every 2 units along the x-axis. A clear understanding of the period allows us to anticipate the wave patterns along different intervals.

Vertical Shift

Vertical shift in trigonometric functions involves shifting the entire graph up or down along the y-axis. This shift effectively changes the baseline from which the waveform reaches its peaks and valleys, but it does not affect the function's amplitude or period. This is determined by the \( D \) in the equation \( y = A \cos(Bx + C) + D \).
In the function \( y = 1 + \frac{1}{2} \cos(\pi x) \):

• The term \( +1 \) indicates a vertical shift of 1 unit upward.
• The original midline \( y = 0 \) becomes \( y = 1 \).

This vertical shift keeps the function's structure the same, just elevated or lowered by a fixed amount. Understanding vertical shifts helps in plotting the graph as it defines the level about which the function oscillates. Knowing this can aid in matching real-life scenarios where a wave or periodic activity occurs around a different baseline.