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    I've been given functions to sketch and I can never figure out how to sketch them.

    Some of the functions include;

    x^2sin(\frac{1}{x})

     |x|^3+|y|^3 = 1

     \frac{1}{x+2} sin(x^{-\frac{3}{2}})

    For the trig types, I usually try to use a Taylor's series, but this rarely helps.
    I usually try to observe what happens as x tends to 0 and infinity or any points near where f tends to +- infinity, also I often try finding the derivative to help with the sketching, but I can never seem to get an accurate sketch and sometimes I have no idea where to start, like the last two above.

    So, is there a general set of things to consider that helps with curve sketching ?
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    (Original post by NotNotBatman)
    I've been given functions to sketch and I can never figure out how to sketch them.

    Some of the functions include;

    x^2sin(\frac{1}{x})

     |x|^3+|y|^3 = 1

     \frac{1}{x+2} sin(x^{-\frac{3}{2}})

    For the trig types, I usually try to use a Taylor's series, but this rarely helps.
    I usually try to observe what happens as x tends to 0 and infinity or any points near where f tends to +- infinity, also I often try finding the derivative to help with the sketching, but I can never seem to get an accurate sketch and sometimes I have no idea where to start, like the last two above.

    So, is there a general set of things to consider that helps with curve sketching ?
    You can probably find a nice list of things to consider by googling, so I won't write my own out. You have mentioned some of them already.

    However, it really comes down to experience and practise. There's no magic bullet and you just have to use your brain a bit (or a lot..)

    e.g. |x|^3+|y|^3=1: note that like the operation (x,y) \to (x^2, y^2), the modulus operator maps (x,y) into the first quadrant. So whatever the function does there (x^3+y^3=1 in this case) is mirrored into the other 3 quadrants, since if, say, (1,1) is in the locus of the curve, then so are (-1,1), (1,-1), and (-1,-1). (or to put it another way: the function is symmetrical in both x and y).

    So you only need to figure out what x^3+y^3=1 looks like in the first quadrant then reflect it. Then you could say, maybe, that in the first quadrant:

    x > 0 \Rightarrow y = \sqrt[3]{1-x^3} < 1 and similarly x < 1

    so we need only consider points in the the unit square.

    Then for small x, y = \sqrt[3]{1-x^3} \approx 1-\frac{x^3}{3} so for x near 0, the curve looks like an easily plotted cubic, and similarly for small y, it'll be cubic, but with x and y swapped.

    And so on.
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    (Original post by NotNotBatman)
    I've been given functions to sketch and I can never figure out how to sketch them.

    Some of the functions include;

    x^2sin(\frac{1}{x})

     |x|^3+|y|^3 = 1

     \frac{1}{x+2} sin(x^{-\frac{3}{2}})

    For the trig types, I usually try to use a Taylor's series, but this rarely helps.
    I usually try to observe what happens as x tends to 0 and infinity or any points near where f tends to +- infinity, also I often try finding the derivative to help with the sketching, but I can never seem to get an accurate sketch and sometimes I have no idea where to start, like the last two above.

    So, is there a general set of things to consider that helps with curve sketching ?
    First, find the asymptotes. These are lines that a graph tends to but never reaches (although be careful - some graphs will cross the asymptote then go back the other way and slowly tend to it). For vertical asymptotes, look for x-values that would result in an undefined output (usually dividing by zero). For horizontal asymptotes, look for the long-term behavior by substituting in very large (both positive and negative) x-values.

    Sometimes you'll have to deal with other asymptotes. For example, the curve (x^2 + 3x + 5) / (x-4) tends to y=x when x is very large. You can spot this by substituting the large values, but to get a proper relationship, look at the largest power of x on the top and bottom (note that a^x where a is a constant will trump any power of x). When x becomes large enough, no other terms matter. So in this case we have x^2 on the top and x on the bottom which makes y=x. Also look at behaviour near asyptotes (for example, substitute in x = 3.999 and x = 4.001 for an asymptote of x=4) to see whether y is positive or negative.

    Then find the x and y intercepts and any turning points (it helps if you can differentiate the function, which should be possible for the first and last examples you gave).

    For the absolute-value one, a tip would be that neither x nor y can be greater than 1, and you only have to look at one quadrant since it would be reflected in all of them.
 
 
 
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