What are some books that illustrate or teach the techniques to sketch the graph of such curves as listed below, analytically or elsewise without manually putting and values for x and then finding out y.
- |x| + |y| = 1
My attempt: We can write this as $|y| = 1 - |x|.$ If $y \geqslant 0,$ $y = 1 - |x|$ and if $y < 0$ then $y = |x| - 1.$
Thus, combining both we get a diamond shaped curve with both horizontal vertices having two intersecting lines each, since $1-|x|$ is the upside down curve of $|x|$ shifted 1 unit up on Cartesian plane and $|x| - 1$ is the regular $|x|$ V curve shifted 1 unit downwards.
Now if one plots $|x| + |y| = 1$ with desmos or any other curve sketching tool, it is quickly evident that what I wrote above isn't correct. Since $|x| + |y| = 1$ is simply a diamond centered at the origin but without any extra arms lying at the horizontal vertices.
- $|y| = \sin (x)$
Similar to the previous one, considering $|y|$ branch wise for $y \geqslant 0$ and $y < 0$ gives two functions and combining them result in a intersecting curve of $- \sin (x)$ and $\sin (x).$
Again, this is incorrect since when $|y| = \sin (x)$ is plot using desmos, the curve has gaps between certain intervals.
- $y = \frac{x^2 + 1}{x^2 - 1}$ and \begin{equation*} f(x) = \begin{cases} \sin\left(\frac{\pi}{4}(x - \lfloor x \rfloor )\right)& \text{if } \lfloor x \rfloor \text{ is odd }, x \geqslant 0, \\ \cos\left(\frac{\pi}{4}(1 - x + \lfloor x \rfloor )\right) & \text{if } \lfloor x \rfloor \text{ is even }, x \geqslant 0. \end{cases} \end{equation*}
I fail to come up with anything for rational functions or such piecewise function.
etc. To add a few more examples $(x+y)^{2} = xy^{2},$ $|y| = \sin(x^2) ,$ $y= e^{x}(2x^{2} - 5x +2),$ $ y = 7|x|-|x|^3.$
So what are some references I can study to tackle such questions ? What is basics behind analytically sketching such curves (without a digital sketching tool of any sorts of course).
If such texts doesn't exist, can I get some insight on where I am going wrong with my attempts ?
I do not have my copy of the book available with me right now, but I think Fuller's Analytic Geometry textbook is just what you need.
Or you might find suitable graphing methods in Love and Rainville's Analytic Geometry textbook.
Both textbooks cover graphical methods to a considerable detail.