Problem 13 - Precalculus

Question

For the following exercises, graph two full periods of each function and state the amplitude, period, and midine. State the maximum and minimum \(y\) -values and their corresponding \(x\) -values on one period for \(x>0 .\) Round answers to two decimal places if necessary. $$ f(x)=4 \cos (\pi x) $$

Step-by-Step Solution

Answer

Amplitude: 4, Period: 2, Midline: y=0. Max value: 4 at x=0, Min value: -4 at x=1.

Step 1: Identify Amplitude

The amplitude of a cosine function in the general form \( a \cos(bx + c) + d \) is \( |a| \). Here, the function is \( f(x) = 4 \cos(\pi x) \), so the amplitude is \( |4| = 4 \).

Step 2: Determine Period

The period of a cosine function is calculated using the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \). For \( f(x) = 4 \cos(\pi x) \), \( b = \pi \). Thus, the period is \( \frac{2\pi}{\pi} = 2 \).

Step 3: Identify Midline

The midline of a cosine function in the form \( a \cos(bx + c) + d \) is \( y = d \). In our function, no additional constant is added, so \( d = 0 \) and the midline is \( y = 0 \).

Step 4: Graph Two Full Periods

For two full periods, use the period \( P = 2 \) identified earlier. Plot \( f(x) \) starting from \( x = 0 \) to \( x = 4 \). Key points for one period are \( x = 0, 0.5, 1.0, 1.5, 2 \), then repeat for another period.

Step 5: Identify Max and Min on One Period for \( x > 0 \)

The maximum value of the function is the amplitude at \( x = 0, 2, 4, \ldots \), so the maximum is \( 4 \) at \( x = 0 \) and \( x = 2 \). The minimum value occurs at \( x = 1, 3, 5, \ldots \), so the minimum is \(-4\) at \( x = 1 \).

Step 6: Summarize Characteristics

The function \( f(x) = 4 \cos(\pi x) \) has an amplitude of 4, a period of 2, and a midline of \( y = 0 \). Maximum value is 4 at \( x = 0 \) and minimum value is -4 at \( x = 1 \). Two full periods span from \( x = -2 \) to \( x = 4 \).

Key Concepts

Understanding Amplitude in Trigonometric Functions

Amplitude in a trigonometric function such as cosine or sine represents the height of the wave's peaks. It signifies how much the function deviates vertically from its midline, indicating the maximum change in value from the midline. For functions in the form \( a \cos(bx + c) + d \), where \( a \) is the amplitude, the amplitude is represented by the absolute value \(|a|\). This ensures it is always a positive number, as it describes a distance or magnitude.

• For the function \( f(x) = 4 \cos(\pi x) \), the amplitude is \(|4| = 4\).
• This value indicates that the peaks of the function are 4 units above and 4 units below the midline.

Understanding amplitude helps in sketching the graph as it determines the vertical stretch or compression of the wave. A larger amplitude means taller waves, while a smaller amplitude results in shorter waves.

The Concept of Period in Trigonometric Functions

The period of a trigonometric function is the interval over which the function repeats itself. It's a crucial feature that informs us of how frequently the cycle occurs within a given domain. For cosine functions, the period can be found using the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \).

• In our function \( f(x) = 4 \cos(\pi x) \), we identify \( b = \pi \).
• The period is calculated as \( \frac{2\pi}{\pi} = 2 \).

Thus, each cycle of the cosine wave occurs over an interval of 2 units along the \( x \)-axis. This unique trait of periodicity means that, after each period, the function will have identical behavior, showing the same maximum, minimum, and crossing points relative to the midline.

Exploring the Midline in Trigonometric Functions

The midline of a trigonometric function is a horizontal line that runs through the middle of the wave, bisecting its graph. It represents the average value of the function, around which the wave oscillates. This is particularly determined by the \( d \) constant in the general form \( a \cos(bx + c) + d \).

• For the function \( f(x) = 4 \cos(\pi x) \), there is no \( d \) term present, meaning \( d = 0 \).
• Thus, the midline is simply \( y = 0 \).

The midline indicates the baseline level of the wave and helps determine the vertical shifts. Knowing the midline allows us to better understand the function's movement in relation to its average or baseline position. In cases where the midline is not \( y = 0 \), the entire graph of the function would shift vertically by the value of \( d \).