Good resources to improve graph sketching skills?

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BP_Tranquility
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Does anyone know of any good books/websites/resources etc that help with improving graph sketching skills and/or provide extra practice for difficult graphs?

In particular, I'm looking for anything that would help in preparing for a maths interview at Cambridge for physical natural sciences

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Tanqueray91
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(Original post by BP_Tranquility)
Does anyone know of any good books/websites/resources etc that help with improving graph sketching skills and/or provide extra practice for difficult graphs?

In particular, I'm looking for anything that would help in preparing for a maths interview at Cambridge for physical natural sciences

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What sort of graphs do you mean. Are you talking about wanting to be able to draw some common graphs from memory?
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http://nrich.maths.org/public/leg.php?code=-140

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BP_Tranquility
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(Original post by mobbsy91)
What sort of graphs do you mean. Are you talking about wanting to be able to draw some common graphs from memory?
Graphs like e^(1/x), tan³x , sinx/x, tan²x/x -- graphs I could be asked to draw at an interview (since more complex graphs like these aren't asked at AS so I don't have much experience on how to go about drawing them beyond finding the asymptote, and seeing if they have stationary points)

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Tanqueray91
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(Original post by BP_Tranquility)
Graphs like e^(1/x), tan³x , sinx/x, tan²x/x -- graphs I could be asked to draw at an interview (since more complex graphs like these aren't asked at AS so I don't have much experience on how to go about drawing them beyond finding the asymptote, and seeing if they have stationary points)

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I understand, the above link the other poster posted is good, and also, if you have a Maths textbook, the graphs that are in there would be a good start (not sure which board you do but Edexcel have the different graphs) - alternatively, ask your teachers for some good graphs to know!
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BP_Tranquility
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For the graph y=sinx/x, I'm confused on how to draw it. I've differentiated it and to find the stationary points, I end up with tanx=x for which x=0, but then finding the value for y gives infinity so there aren't any stationary points? But for values of x= pi, 2pi, y=0 so it's a periodic function so there must be stationary points
Also, isn't x=0 an asymptote.

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(Original post by BP_Tranquility)
For the graph y=sinx/x, I'm confused on how to draw it. I've differentiated it and to find the stationary points, I end up with tanx=x for which x=0, but then finding the value for y gives infinity so there aren't any stationary points? But for values of x= pi, 2pi, y=0 so it's a periodic function so there must be stationary points
Also, isn't x=0 an asymptote.

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lim x --> + infty = 0
lim x --> - infty = 0

function isn't defined at x=0 but lim x --> 0 = 1 (too long to cover here but you might know it)

yes, its equal to 0 whenever the numerator is equal to 0 but that doesn't mean its periodic

not quite, there are lots of stationary points

think of the graph of y=x and y=tan(x) and where they intersect, the exact values are hard to find but you should see there are quite a few

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BP_Tranquility
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Thanks

Also, how do you go about drawing something like y=sin(x²)? As in what steps do you follow?

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atsruser
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(Original post by BP_Tranquility)
Also, how do you go about drawing something like y=sin(x²)? As in what steps do you follow?
There's more than one way to approach it, but to visualise what happens note that:

a) the graph of y=x^2 compresses points on the y-axis, in a non-linear fashion, to points on the x-axis; note that the x^2 term represents the height on the y-axis.

So, for example, the range [0,1] on the y-axis maps to [0,1] on the x-axis but the range [1,4] on the y-axis maps to [1,2] on the x-axis.

This means that (since we're feeding x^2 into \sin()) that all of the variation of \sin between values 1 and 4 take place on the x-axis between values 1 and 2.

This effect increases the further up the y-axis we go; the range 4-9 on the y-axis is compressed to the range 2-3 on the x-axis.

In the case of \sin(x^2), it should be clear that the frequency of the function will gradually increase as x increases, due to this effect - we will see more and more cycles compressed into the same region of the x-axis.

b) \sin(x^2) is an even function; i.e. \sin(-x)^2=\sin(x^2) so it is symmetrical about the y-axis. That means that we only need to figure out what it does for +ve values of x, then draw a mirror image of that part of the function to find what it does for -ve values of x.
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BP_Tranquility
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Thanks

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WhiteGroupMaths
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(Original post by BP_Tranquility)
Does anyone know of any good books/websites/resources etc that help with improving graph sketching skills and/or provide extra practice for difficult graphs?

In particular, I'm looking for anything that would help in preparing for a maths interview at Cambridge for physical natural sciences

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I wrote some stuff on a supplementary website, hopefully you might find them useful:

Quick Graphing 1


Quick Graphing 2

Quick Graphing 3

All the best for your upcoming interview. Peace.
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Mike_123
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(Original post by BP_Tranquility)
Thanks

Also, how do you go about drawing something like y=sin(x²)? As in what steps do you follow?

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Consider the domain, intercepts, symmetry (odd/even), asymptotes, intervals of increase/decrease (f'(x)), local min/max, concavity and points of inflection
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BP_Tranquility
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(Original post by WhiteGroupMaths)
I wrote some stuff on a supplementary website, hopefully you might find them useful:

Quick Graphing 1


Quick Graphing 2

Quick Graphing 3

All the best for your upcoming interview. Peace.
Thanks, will have a look

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BP_Tranquility
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(Original post by Mike_123)
Consider the domain, intercepts, symmetry (odd/even), asymptotes, intervals of increase/decrease (f'(x)), local min/max, concavity and points of inflection
Cheers- don't suppose you could elaborate on finding the concavity and symmetry of the function please?

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Mike_123
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(Original post by BP_Tranquility)
Cheers- don't suppose you could elaborate on finding the concavity and symmetry of the function please?
c) Symmetry: Determine whether the function is an odd function, an even function or neither odd nor even. If f(-x) = f(x) for all x in the domain, then f is even and symmetric about the y-axis. If f(-x) = -f(x) for all x in the domain, then f is odd and symmetric about the origin.

g) Concavity and Points of Inflection : We must determine when f''(x) is positive and negative to find the intervals where the function is concave upward and concave downward. Inflection points occur whenever the curve changes in concavity.


Good luck in your interview.
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BP_Tranquility
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Thanks
I don't suppose you know of any good websites/documents etc where I could get a list of 'difficult' graphs to practice drawing?

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BP_Tranquility
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I've just learned about the second derivative test to check for concavity- can it be used for any function or just polynomials? I've tried f(x)=e^(-0.5x² ) and f''(-2) is returned as being concave up even though the curve is tending towards the y=0 asymptote

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