I've been trying to learn how to graph some miscellaneous functions and I came across this. I put it in Desmos and this is what I ended up with.
Can you let me know how did the graph get its shape? And how can one graph such type of functions on their own?
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The input values for the graph are measured in radians by default. Between 0 and $\pi$ radians, the sine function's outputs have positive values, so when added to the number of radians produces values above the line y = x. Between $\pi$ and $2\pi$ radians, the sine function has negative outputs which when added to the number of radians produces values below the line y = x. This pattern then repeats around the line y = x in each horizontal direction, just like the sine function itself does.
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The graph for $y = x + \sin x$ is the superposition of the lines $y = x$ and the sine curve $y = \sin x$. When you combine one graph and the other through a mathematical operation, the new graph takes on the characteristics of each of the original graphs.
In your Desmos graph, $y = x + \sin x$ can be thought of as the regular sine function $y = \sin x$ rotated $45°$ clockwise.
The function has two parts.
The $x$ is a line at 45° and it is shown as a dashed line in your post.
The $\sin x$ is a sine wave that fluctuates around the 0 value by $\pm 1$ in a wavy pattern.
Put them together, and you get a curve going uphill but also fluctuates by $\pm 1$ in a wavy pattern.