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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.

Matthew Towers
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Romin
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  • I think it would help if you have some examples for us, for the second paragraph. For the third paragraph, solving by inspection does crop up a lot... if you mash together some arbitrary functions and symbols you can very quickly create equations which are basically impossible to solve using any particular analytic method. So, the only solution is the ‘obvious’ one you get by looking at it. A common theme in contest problems, or textbook exercises, is to show that the ‘obvious’ solution is the only solution. – FShrike Aug 27 '22 at 07:24
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    «how should I graph that equation? » I think a preliminary step could be to use one of the many graphing tools available online, such as https://www.desmos.com/calculator – Gribouillis Aug 27 '22 at 07:31
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    Unless you have graphed an equation before, or can identify properties by observation of its components (product of an exponential and sine, looks exactly as one should expect, if one knows what those words mean) you really need to try plotted some of them. Manually if they are simple enough, but using software if you want to play dynamically with any coefficients and constants. I prefer Geogebra to Desmos, but do what you like. – Nij Aug 27 '22 at 07:54
  • @FShrike I've added an example for the second paragraph. And thank you for the answer to third paragraph. I would like to ask another question actually, which is, since during the learning phase it's hard to know if what one has grasped from a topic is correct or not, can this stack exchange be used to clarify those knowledge? I'm kind of asking regarding the guideline for it. It would be really helpful to know. – Romin Aug 27 '22 at 11:05
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    @Romin This StackExchange is best suited to precise questions. Unfortunately you want to have a super-general guide to graphing equations, or the loci of equations, and - in general - this is very hard. If you reduced your question to something more concrete, e.g. “how can I use first and second derivatives to sketch rational functions?” (And if you added some examples and attempts of your own) Then that would work well and you’d get good help. – FShrike Aug 27 '22 at 11:19
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    @FShrike Yeah I understand. Alright, I will keep that in mind and try to create better questions. Thank you for the help! – Romin Aug 27 '22 at 11:41
  • My answer is: "Experience with a wide variety of functions". As asked, your question is on the level with "How do I repair a car?" – JonathanZ Aug 27 '22 at 13:55
  • My comment-reply to a similar question from yesterday: – ryang Aug 28 '22 at 14:31
  • Graphing a complicated function is just about investigating it then piecing together various clues, and if there are still gaps, investigating further. By investigation, I mean to to ask questions like what the intercepts are, what happens as x gets very big, and then very small, what does it mean when an x-value makes the denominator 0, how does the curve's gradient vary, etc. Eventually, you will attain a better sense of how common functions look like, and any required investigation will speed up significantly. – ryang Aug 28 '22 at 14:31

1 Answers1

5

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.]

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.

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.

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).

VoidGawd
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