General Strategy For Sketching The Graphs Of Unusual Equations

Question

To give this question some context. I am a grade 11 student who is going through the equivalent of a Precalculus course. I have, however saw questions asking for sketches of curves with some very strange equations, such as:

• $\sin(y)=\sin(x)$
• $(x^8+4yx^6+6y^2x^4+4y^3x^2+y^4)^2=1$
• Or even this:

So I'm curious. Is there are general strategy of plotting out these curves, without simply calculating the x and y values of every point? If so, what is it? Thanks in advance.

Answer 1

Not sure that there is a common strategy. But here is what I would do in some of these cases:

• $\sin y=\sin x$: Obviously $y=x$ is a solution. But due to the periodicity of the $\sin$ function, the answer is $y=x+2k\pi$, with $k\in \mathbb Z$. That's an infinite set of parallel lines to $y=x$
• $(x^8+4yx^6+6y^2x^4+4y^3x^2+y^4)^2=1$: for this you need to notice that you can write it as $(x^2+y)^8=1$. For real numbers, you have $y+x^2=\pm 1$ or $y=\pm 1-x^2$. These are two parabolas shifted along the $y$ axis.
• for the third case, if we talk about the same polynomial $p$ then $p(y)=-p(x)$, so it means that it's symmetric with respect to $y=-x$. Only the last two obey this requirement. For the circle $p(x)=x^2-r^2/2$, so $p(x)+p(y)=0$ means $x^2+y^2=r^2$. For the line $p(x)=x$ so $y=-x$.