How To Get An Oxford Engineering Offer: Bonus Chapter - Graph SketchingWatch
AKA: Oxbridge interview style graph sketching for STEM students in 5 simple-ish steps:
Sketching unusual graphs is a key skill and requires some strategies. If any of these factors prove very awkward to find or, if during an interview, you are advised not to look at this aspect yet carry on moving down the list. After drawing and labelling your axes, here’s the method I follow:
- Inspection. Certain graphs may be in a standard form you already know, like a circle or an exponential curve, which can simplify the drawing process. Additionally, in some problems you can tell if a function is symmetrical by inspection, for example even functions like x^2 sometimes lead to vertical lines of symmetry. This means you can sketch one side of the graph then reflect this to form the other side, saving time.
- Asymptotes. For vertical asymptotes, find any values where f(x) may not exist on the x-axis – for example, on the graph 1/x the function is undefined at x=0. Some functions have a restricted domain, such as ln(x) which exists for x > 0, which may inform any asymptotes. For horizontal asymptotes, consider occurs as x tends towards infinity. Does the function tend towards infinity, 0 or another value? Lastly, some functions may have “oblique” asymptotes; to find them divide the function’s numerator by the denominator and consider the behaviour as x tends to infinity. Limits may be required to test what happens as x tends to 0, another undefined value, or infinity; the L’Hopital rule can be very handy for this.
- x and y intercepts. Substitute x=0 and y=0 into your function and solve for the other variable to find these points.
- Derivative and turning points. Finding stationary points allows you to showcase your differentiation skills to an interviewer and is essential to accurately plot the graph. Sometimes whether they are maxima or minima can be observed by testing points around them and using inspection, however using the second derivative to prove their nature (whether f’’(x) is >, < or = to 0) is more rigorous. The second derivative can too be used to find points of inflection; however, function sketching the exact value of these is often less important to the general graph.
- Test some values. Be systematic in this and don’t just choose random values. Testing values can help you get an idea of the exact shape of the graph (e.g. is the initial increase sudden or slow?).
Here’s some good examples of graphs to practice sketching. Check your answer using Desmos (https://www.desmos.com/):
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