Problem 56 Graph the given function by usin... [FREE SOLUTION]

Chapter 4: Problem 56

Graph the given function by using the addition-of-ordinates method. $$y=\frac{2 x}{3}-\sin x$$

Short Answer

Expert verified

Using the addition-of-ordinates method, each graphed separately, the linear function would yield a straight line, and the sinusoidal correlation would give an inverted sine wave. When combined by adding the y-values of the two, it results in a sinusoidally modulated straight line for the function $y=\frac{2x}{3}-\sin x$.

Step by step solution

Plot The Linear Function

Start by plotting the linear function $y=\frac{2x}{3}$. Choose values for x and calculate the corresponding y-values. For example, you might choose x-values from -10 to 10. Remember that the graph of this function is a straight line with a slope of 2/3.

Plot The Sinusoidal Function

Next, plot the function $y=-\sin x$. Agree on the same set of x-values to maintain consistency. The y-values will range from -1 to 1 as the sine function oscillates between these values. Remember, the minus sign will flip the standard sine graph about the x-axis, making the wave's peaks become troughs and vice versa.

Combine the two graphs with the addition-of-ordinates method

Finally, combine the two graphs using the addition-of-ordinates method. For each x-value, add the y-value from the linear function to the y-value from the sinusoidal function. This will yield the y-value for the combined function $y=\frac{2x}{3}-\sin x$. Graph these points and draw a smooth curve through them to get the final graph of the function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Addition of Ordinates

When graphing a function that is a combination of two simpler functions, we can use the addition of ordinates method. Consider two functions plotted on the same set of axes. The ordinate refers to the y-value at each point on the graph. The addition of ordinates method involves adding the y-values (ordinates) of these two functions for each corresponding x-value.

First, calculate the y-values for each function separately at the same x-values.
Then, simply add these y-values together to find the ordinance of the new combined function.

The result gives you the new y-coordinate for the combined graph, which you can then plot to visualize the overall function.

Linear Functions

A linear function is one of the simplest types of functions, characterized by a straight line on a graph. This type of function can be expressed in the form of $y = mx + c$, where $m$ is the slope, and $c$ is the y-intercept.

Several key properties of linear functions include:

The slope $m$ shows the rate of change or how steep the line is. A positive slope indicates that the line rises as it moves to the right.
The y-intercept $c$ is where the line crosses the y-axis.

In the exercise, the linear function $y = \frac{2x}{3}$ has a slope of $\frac{2}{3}$, telling us that for every unit increase in $x$, $y$ increases by $\frac{2}{3}$. The plot will therefore be a steady, rising line.

Sinusoidal Functions

Sinusoidal functions are characterized by their wave-like pattern. A basic sine function, $y = \sin x$, oscillates between -1 and 1. These functions are periodic, meaning they repeat after a certain interval, known as the period.

For $y = -\sin x$ in the exercise:

The negative sign flips the standard sine wave across the x-axis, turning peaks into troughs and vice versa.
The amplitude of 1 indicates how high and low the waves reach.
The period of the sine function is $2\pi$, meaning it completes a full cycle every $2\pi$ units.

Understanding these characteristics helps in plotting and combining it with other functions.

Combining Graphs

Combining graphs involves overlaying two or more separate functions onto the same plot to see how they interact. Using the addition of ordinates, you'll create a more complex graph that embodies characteristics from each of the original functions.

Steps to combine graphs:

First, ensure both functions are plotted with the same scale and range of x-values.
For each x-value, add the y-value from the linear function to the y-value from the sinusoidal function.
Plot these new y-values to construct the graph of the combined function.

This method helps reveal how different functions influence the overall behavior of a system, capturing elements such as periodic oscillations overlaying linear growth.

91影视

Graph the given function by using the addition-of-ordinates method. $$y=\frac{2 x}{3}-\sin x$$

Short Answer

Step by step solution

Plot The Linear Function

Plot The Sinusoidal Function

Combine the two graphs with the addition-of-ordinates method

Key Concepts

Addition of Ordinates

Linear Functions

Sinusoidal Functions

Combining Graphs

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