Problem 69 - Precalculus Essentials

Question

Graph one period of each function. $$y=-|3 \sin \pi x|$$

Step-by-Step Solution

Answer

The graph of one period of the function y=-|3sin(πx)| consists of two downward-pointing peaks at \(x=\frac{1}{2}\) and \(x=\frac{3}{2}\), each with a minimum value of -3, and two x-intercepts at x=0 and x=2.

Step 1: Identify the Transformations

The function is \(y=-|3 \sin \pi x|\), which is a transformation of the absolute value of sine function. Here, the absolute value would make all negative y-values of the function positive, and the negative sign in front of the absolute value will invert the positive part. The coefficient 3 inside the absolute value changes amplitude of the sin function, while the multiplier π to the x modulates the period of the function. A positive coefficient in front of x inside the sine function signifies that the function will move faster, hence, decreasing its period from 2π to 2.

Step 2: Determine the Key Points of the Original Sine Function

Start by plotting a base graph of one period of the original function sin(x). The key points occur within each period at multiples of \(\frac{\pi}{2}\), starting from 0 at these points: \(x=0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\), and \(2\pi\), sin(x) values are 0, 1, 0, -1, and 0 respectively.

Step 3: Adjust for Amplitude

The absolute value of the coefficient 3 changes the amplitude of the sin function. As a result, each y-coordinate of the key points should be multiplied by 3, which will result in values of 0, 3, 0, -3, and 0.

Step 4: Adjust for Period

Since the function is multiplied by π, all x-coordinates of the key points will be divided by π, resulting in points: \(x=0, \frac{1}{2}, 1, \frac{3}{2}\), and 2.

Step 5: Apply the Absolute Value

Apply the absolute value transformation. After this transformation all y values are positive, so points are \(x=0, \frac{1}{2}, 1, \frac{3}{2}\), and 2, with corresponding y values 0, 3, 0, 3, and 0.

Step 6: Reflect Across the X-axis

The negative sign in front of the absolute value inverts the graph over the x-axis. After the translation, the final key points we have are: at \(x=0, \frac{1}{2}, 1, \frac{3}{2}\), and 2, y values are 0, -3, 0, -3, and 0.

Step 7: Sketch the Graph

Now plot these coordinates onto a graph to get a visual representation of one period of the function. This function repeats this pattern for every period.