Problem 86 - Precalculus

Question

In Exercises \(67-94,\) add the ordinates of the individual functions to graph each summed function on the indicated interval. $$y=-\sin \left(\frac{\pi}{4} x\right)-3 \sin \left(\frac{5 \pi}{4} x\right), 0 \leq x \leq 4$$

Step-by-Step Solution

Answer

Sum and plot the ordinates for given functions within the interval to get the graph.

Step 1: Understand the Functions

We have two functions to consider: \( y_1 = -\sin\left(\frac{\pi}{4} x\right) \) and \( y_2 = -3\sin\left(\frac{5\pi}{4} x\right) \). Our goal is to sum these functions within the interval \( 0 \leq x \leq 4 \).

Step 2: Compute the Ordinates (Sum of Functions)

For each value of \( x \) within the interval \( 0 \leq x \leq 4 \), we need to calculate the sum of the ordinates. That is, for a given \( x \), compute \( y = y_1 + y_2 = -\sin\left(\frac{\pi}{4} x\right) - 3\sin\left(\frac{5\pi}{4} x\right) \).

Step 3: Plot Points

For several values of \( x \) (such as 0, 1, 2, 3, 4, etc.), calculate \( y = -\sin\left(\frac{\pi}{4} x\right) - 3\sin\left(\frac{5\pi}{4} x\right) \) and plot these points on a coordinate plane.

Step 4: Draw the Summed Graph

Connect the plotted points to graph the summed function over the interval \( 0 \leq x \leq 4 \). Ensure smooth curves by calculating more points if necessary.

Key Concepts

Ordinate Calculation

Calculating ordinates is a crucial step in graphing trigonometric functions, especially when dealing with the sum of multiple functions. The ordinates are simply the y-values of a function corresponding to given x-values. For each x-value, you calculate the y-value by substituting x into the given function. In this exercise, we are working with the equation:
\( y = -\sin\left(\frac{\pi}{4} x\right) - 3\sin\left(\frac{5\pi}{4} x\right) \)

• This equation is a combination of two sine functions.
• The first function, \( y_1 = -\sin\left(\frac{\pi}{4} x\right) \), is a scaled and flipped sine function with a specific frequency.
• The second function, \( y_2 = -3\sin\left(\frac{5\pi}{4} x\right) \), is also scaled but involves a larger magnitude and frequency shift.

For ordinate calculation, simply calculate \( y_1 \) and \( y_2 \) for each x given in the interval \(0 \leq x \leq 4\), then sum the two results to obtain the ordinate for each x. This process gives you the points needed to graph the combined function.

Sine Function

Understanding the sine function is essential for accurately working with trigonometric graphs. The standard sine function is expressed as:
\( y = \sin(x) \)
The sine function oscillates between -1 and 1, and it has a period of \( 2\pi \), meaning it repeats its pattern every \( 2\pi \) units along the x-axis.
When transformed (as in this exercise), the sine function can change in the following ways:

• Amplitude: The coefficient in front changes the height of the wave.
• Frequency: The frequency is altered by multiplying x inside the function, affecting how many waves occur in a given interval.
• Phase Shift: A term added or subtracted inside the sine function, not present here, could shift the wave left or right.

In our exercise, two sine functions are adjusted in amplitude and frequency, affecting how steeply they rise or fall and the number of cycles between 0 and 4. By understanding these transformations, graphing becomes more intuitive.

Plotting Coordinates

Plotting the calculated coordinates on a graph is where the visualization of the trigonometric function summation happens. For each computed ordinate (y-value), plot the corresponding point on the graph at each x-value.
These steps break the process down:

• Select x-values systematically within the interval (e.g., 0, 1, 2, 3, 4).
• Calculate the y-value for each x using the summed function.
• Plot each point \((x, y)\) on the coordinate plane.

After placing all the points, you will need to visually connect them to see the complete wave pattern. This is done by drawing a smooth curve through the points. The more points you calculate and plot, the smoother and more accurate your graph will be.
By integrating these detailed steps of plotting coordinates, the summation of the trigonometric functions becomes visible, revealing the overall behaviour of the combined trigonometric graph. This visualization solidifies the concept and makes the math come to life visually.