Problem 69 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graph of the function \(y = -|3sinπx|\) is a wave pattern that varies between -3 and 0 due to the modifications from the absolute value and negative sign. It is identical to the graph of the sine function for half its period, but mirrored above the x-axis for the other half. One complete period for this function occurs from x = -1 to x = 1.

Step 1: Graph the original function

First, graph \(y = 3sinπx\). The sine function has a maximum value of 1 and a minimum value of -1, which means there will be a maximum at 3 and a minimum at -3. The value -sinx is only negative in the second and fourth quadrants. Accordingly, the graph of y=-sinx is going to have a positive value in these quadrants.

Step 2: Apply the absolute value

Apply the absolute value function to the original sin function: \(y = |3sinπx|\). Absolute value is a function which makes any negative number positive. So in the graph, any part of the sine graph below the x-axis will get flipped above.

Step 3: Apply the reflection

Finally, apply the negative sign to obtain \(y = -|3sinπx|\). The negative sign before the absolute value function indicates a reflection over the x-axis. This flips the graph from step 2 over the x-axis.

Step 4: Identify one period

Identify one period of the function. For the function \(y = -|3sinπx|\), the period is the same as the regular sine function, due to the coefficient π in front of x. Therefore, one period is -1 <= x <= 1.