Problem 27 - Precalculus Mathematics for Calculus

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; Vertical shift: 1 unit up.

Step 1: Identify the amplitude of the cosine function

The amplitude of the function \( y = 1 + \frac{1}{2} \cos(\pi x) \) is determined by the coefficient of the cosine term. Here, the coefficient of \( \cos(\pi x) \) is \( \frac{1}{2} \). So, the amplitude is \( \frac{1}{2} \).

Step 2: Determine the period of the cosine function

The period of a cosine function \( \cos(bx) \) is given by \( \frac{2\pi}{b} \). In this case, \( b = \pi \), so the period of the function is \( \frac{2\pi}{\pi} = 2 \).

Step 3: Describe the vertical shift of the function

The function \( y = 1 + \frac{1}{2} \cos(\pi x) \) includes a constant term \( +1 \), which shifts the entire graph vertically upward by 1 unit.

Step 4: Sketch the graph of the function

To sketch the graph, start with a standard cosine wave. Use the amplitude to determine the maximum and minimum values, which are \( 1 + \frac{1}{2} = 1.5 \) and \( 1 - \frac{1}{2} = 0.5 \), respectively. The period is 2, so the cycle repeats every 2 units along the x-axis. Plot key points at the beginning, quarter, halfway, three-quarters, and end of the period, then shift the wave vertically by 1 unit.

Key Concepts

Amplitude

When looking at a trigonometric function like our cosine example, amplitude is a key property that helps us understand the graph's vertical stretch or compression. Amplitude is the height from the middle line of the wave to its peak or trough. It is always a positive number and is represented by the coefficient of the \cos \(or \sin\) function, excluding any constants that may appear in front of the function.
In the exercise solution, given that the function is \(y = 1 + \frac{1}{2} \cos(\pi x)\), the amplitude is represented by the \frac{1}{2}\, which is the coefficient of \cos(\pi x)\. Thus, we conclude that the amplitude is \frac{1}{2}\. This tells us how far the peaks rise and the troughs fall from the function's midline. Remember, the midline is not always at \y = 0\; in this case, it's shifted to \y = 1\ because of the vertical shift.

Period

The period of a trigonometric function tells us how long it takes for the function to complete one full cycle. For \cos(bx)\ and \sin(bx)\ functions, the period is given by the formula \frac{2\pi}{b}\.
In our problem, the function is \(y = 1 + \frac{1}{2} \cos(\pi x)\), so we identify \pi\ as \b\. Thus, to find the period, we calculate \frac{2\pi}{\pi} = 2\. This tells us that the graph completes a cycle every 2 units along the x-axis.

• This knowledge helps in sketching because:
• It indicates where key points of the function will repeat.
• Enables plotting of start and end points for the curve on the graph.

Knowing the period is essential for understanding how frequently the graph repeats. Remember, any transformations to the function, like horizontal stretches or compressions, directly affect the period.

Graph Sketching

Graph sketching is an important skill that allows us to visualize the behavior of trigonometric functions. To sketch \(y = 1 + \frac{1}{2} \cos(\pi x)\), we piece together information about the function's amplitude, period, and any vertical or horizontal shifts.
Begin your sketch by identifying key features:

• Amplitude, which affects the maximum and minimum values of the curve.
• Period, which dictates how often the function repeats along the x-axis.
• A vertical shift, here \+1\, moving the baseline of oscillation upward.

For our function, the maximum value of \1.5\ is \1 + \frac{1}{2}\ and the minimum value of \0.5\ is \1 - \frac{1}{2}\. These extremities occur due to the graph's amplitude. The period of 2 indicates that you will place these peaks and troughs within an interval of 2 units.
Finally, to enhance your sketch:

• Start plotting points at important intervals like 0, \frac{1}{4}\, \frac{1}{2}\, \frac{3}{4}\, and 1 period.
• Apply the vertical shift to all points.
• Connect these points smoothly to show the cosine wave within 2 units.

After sketching a primary cycle, you can easily extend it along the axis by repeating the same pattern. This complete approach gives a visual representation of all properties in harmony.