# Graph sketching

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#1
If asked in an interview to sketch a graph similar like this would it be worth saying any observations that I make, for example in the question I attached I noted even and odd function even though it wasn’t necessary in actually sketching the graph. Also when approaching this question what are the main hints that give away the shape of the graph for this case. Initially I looked at their points of intersection but realised it wasn’t useful then considered how they behaviour between the region 0 to 2pi since it’s periodic.
Last edited by The A.G; 1 month ago
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1 month ago
#2
(Original post by The A.G)
If asked in an interview to sketch a graph similar like this would it be worth saying any observations that I make, for example in the question I attached I noted even and odd function even though it wasn’t necessary in actually sketching the graph. Also when approaching this question what are the main hints that give away the shape of the graph for this case. Initially I looked at their points of intersection but realised it wasn’t useful then considered how they behaviour between the region 0 to 2pi since it’s periodic.
To my mind it's a useful thing to do - adding observations - since you're demonstrating knowledge and thought processes.

Note that the relationship permits y<0.
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1 month ago
#3
Also, a very simplistic thing to do is to reason about what happens ((x,y) points) when the trig functions are {-1,0,1}, then join the appropriate dots. I think you just do it partially for 1?
Quick and relatively easy.
Last edited by mqb2766; 1 month ago
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#4
(Original post by ghostwalker)
To my mind it's a useful thing to do - adding observations - since you're demonstrating knowledge and thought processes.

Note that the relationship permits y<0.
My bad I forgot to consider the domain of y since cos is even function it’ll have reflectional symmetry about x-axis,thanks
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#5
(Original post by mqb2766)
Also, a very simplistic thing to do is to reason about what happens ((x,y) points) when the trig functions are {-1,0,1}, then join the appropriate dots. I think you just do it partially for 1?
Quick and relatively easy.
I only did it for 0,1. I stated the domain of cos^2(y) >=0 so didn’t need to consider when they =-1 since it doesn’t exist?
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1 month ago
#6
(Original post by The A.G)
I only did it for 0,1. I stated the domain of cos^2(y) >=0 so didn’t need to consider when they =-1 since it doesn’t exist?
The -1,0,1 was for the basic trig function sin, cos, not the square. It helps to reason about valid solutions. Slighlty simpler sketches to do would be
sin(x) = cos(y)
https://www.desmos.com/calculator/oiplwr3g7x
sin^2(x) = cos^2(y)

The only missing regions for yours are when sin(x)<0. which are your vertical strips of width pi. cos(y)<0 correspond to horizontal strips of width pi but they contain the lines. There is no difference for y>0 and y<0 as ghostwalker noted as they're square/even.

For (trig) sketching at this level, odd/even is good to spot but sticking simple numbers in is generally more than sufficient. As soon as you saw a square, you should think of two (+/-) solutions.
Last edited by mqb2766; 1 month ago
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#7
(Original post by mqb2766)
Youve realized, cos(y) = -1, so cos^2(y) = 1. The only missing regions are when sin(x)<0. which are your vertical strips of width pi. cos(y)<0 correspond to horizontal strips. There is no difference for y>0 and y<0 as ghostwalker noted.

For sketching at this level, odd/even is good to spot but sticking simple numbers in is generally more than sufficient. As soon as you saw a square, you should think of two (+/-) solutions.
My bad I meant the range of cos^2(y) but yes I didn’t take into consideration of when cos(y) <0 thanks. Also for graph sketching I haven’t come across an function where differentiating would be useful to pinpoint any global maximum/minimum. But I should I always keep it in mind in case there’s a situation where differentiating is a more effective approach to see the behaviour of a graph
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1 month ago
#8
(Original post by The A.G)
My bad I meant the range of cos^2(y) but yes I didn’t take into consideration of when cos(y) <0 thanks. Also for graph sketching I haven’t come across an function where differentiating would be useful to pinpoint any global maximum/minimum. But I should I always keep it in mind in case there’s a situation where differentiating is a more effective approach to see the behaviour of a graph
I was editing when you replied, so it may make a bit more sense now.

Stationary points are certainly important for sketching some graphs. They get local turning points (and inflection), but you need extra info to reason about global min/max. The -1,+1 here are the periodic min/max values of the trig terms, but obviously differentiation is not necessary here. So yes, differentation should be in your toolbox, though factorization (roots) or completing the square are alternatives/complementary for simple polynomials.

Thinking about asymptotic behaviour sign, vertical asymptotes, horizontal asymptotes, are sometimes important.
Last edited by mqb2766; 1 month ago
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