What is the intuitive way to graph a given equation?

Question

I just recently started to learn some math on my own. It's been fun to be honest. But I am a little bit confused regarding graphing a given equation. In most of the videos and articles I have read, they first identify what the graph would be like, from their previous knowledge base. Like if its a "linear equation", then the graph would be a straight line. Or if it's a quadratic, maybe it's a ellipse or something.

But if I were to approach an equation and assume I don't know what the outcome might be, how should I graph that equation? Will I have to find the possible domain of the equation and figure out the range by using the possible domain values? For example, here's an equation: x² + y² = 100

And as for the method of identifying equations by just looking at the equation structure, do we use this method because that's the result we get most of the time and because of it's fast usage?

I would be really grateful for some answer on this. I am just a bit confused and don't know who to turn to other than online communities right now.

Answer 1

Following sequence if generally followed on the learning curve:

1. Knowing the function: It's Domain and Range.
2. Plotting the graph.
3. Doing so for many more different functions. ($\log x, |x|, \{x\}, x, x^{2n},x^{2n-1}$, etc.)
4. Learning to make changes in graph for some variations in the functions - Graph Transformations.
5. For their application, you're naturally supposed to learn their general form to save time.
6. Application of graphs to solve otherwise complex functions.
7. Knowing their limitations. [Not really useful beyond $3$D or $3$ variables, etc.]

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For point $5$ it's quite normal related to anything else too, to learn the basic forms and memorize/understand the important ones.

This is especially useful when it comes to Conic Sections. Otherwise you'll find yourself unnecessarily wasting your time over the trivial.

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For your example, $x^2+y^2=100$:

1. You may plot many points to finally get to know it indeed is a circle.
2. As locus of points s.t. distance from origin is always constant ($=10$), thus a circle [$\sqrt{(0-x)^2+(0-y)^2}=100$].
3. You know that any equation of form $(x-h)^2+(y-k)^2=r^2$ is equation of circle where Centre: $(h,k)$ and Radius: $r$
4. And even better knowing the even general form of the circle: $x^2+y^2+2hxy+2gx+2fy+c=0$.

The difference is same as in solving $12\times37$:

1. Add $12$, $37$ times.
2. Add $37$, $12$ times.
3. Multiply $12$ by $37$.
4. Doing $370\times10+74$.

Perhaps my analogy isn't the best but hope you get the point.

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Also, this is a good example of the application of graphs over other methods. (Disregard the fact there I'm looking for something else too).