Curve Sketching
Curve sketching is a common topic for interview questions. But this doesn't mean that you will definitely be asked a curve sketching question at interview!
Some things to think about when sketching curves:
- Symmetry can be very powerful! Think about odd/even functions or if you can exchange $x$ and $y$ without changing the equation.
- Consider where the curve crosses axes
- Consider the asymptotic behaviour (behaviour as $x$ and/or $y$ gets very large). Note that a curve can cross an asymptote in some cases!
- Differentiation can be useful to find stationary points, but also to think about the shape of a curve as it approaches a particular point.
- Trying a few numbers to get a feel for the shape of the graph can be extremely useful!
In this video we discuss some curve sketching techniques.
Interview question examples
Here are some questions for you to try. Each one has a link to a video where we discuss the question and some possible methods which can be used to solve it. Make sure you try each question before looking at the video!
Question 1: Sketch the curve $y^2 = 1-|x|$. A rectangle with sides parallel to the $x$ and $y$ axes is drawn inside this curve. What is the largest possible area of the rectangle?
This question was taken from 1989 STEP 1 Question 7. In an interview situation you would only be asked to sketch the curve at first, and then if there was sufficient time left you would be asked the rest of the question. In the video below we discuss the question and how you might tackle it.
Question 2: Sketch the graphs $y = \ln x$ and $y=mx^2$ on the same axes. How many points of intersection do your graphs have? What happens as $m$ changes?
Once you have done these parts of the question, click the button below to reveal the rest of the question.
Rest of question 2
For what value of $m$ do the curves "touch"? ("touch" means that the two curves are tangential at the point where they intersect). If $m$ is very small what can you say about the point or points of intersection of the two curves?
In the video below, one of our undergraduate helpers tackles this question as part of a mock interview.
Question 3: Part (a) Find all the solutions to the equation $x-1=(e-1) \ln x$. Part (b) Sketch $f(x) = e^x-x^e$.
Hint for part (a)
There are a couple of "obvious" solutions to $x-1=(e-1) \ln x$. Can you find these and show that there are no more possible solutions?
In the video below one of our undergraduates tries this problem. The approach for part (a) was not entirely what I expected (but is valid!).
Question 4: Sketch $\sqrt{y^2-1}=\frac 1 2(e^x-e^{-x})$. Sketch $\sqrt{y^2+1}=\frac 1 2(e^x+e^{-x})$.
In the video below one of our undergraduates sketches these graphs. You can also try sketching the other two in the family, i.e. $\sqrt{y^2+1}=\frac 1 2(e^x-e^{-x})$ and $\sqrt{y^2-1}=\frac 1 2(e^x+e^{-x})$.
More practice questions
Here are some more questions which involve sketching graphs. You can use Desmos to check your sketch (but only after you have attempted to sketch it yourself!)
Sketch $y=x^{\frac 1 x}$, where $x>0$. This is a good example of a graph where trying a few values out can be very helpful. I would suggest starting with $x=1$, $x=\frac 1 {10}$ and maybe $x=100$. Here is a video showing a solution.
Sketch $y=(1+x^2)e^x$. Here is a video showing a solution.
Sketch $y(y-1)=x(x-1)$. Here is a video showing a solution. Here is another video showing a different approach.
Sketch $y^2=x^3-x^4$. Here is a video showing a solution.
Sketch $x^3-y^3=1$. Here is a video showing a solution.
Sketch $y=\sin(e^{-x})$
Sketch $\sin^2 x + \cos^2 y = 1$
Sketch $y^2=\dfrac x{(x-1)^2}$
Sketch $y=\dfrac{\sin x}x$ and $y=\dfrac{\sin x}{x-1}$
1989 STEP 1 Question 7. This is the question used in example 1 above, but with an additional part.
1988 STEP 1 Question 1. This is a nice example of using a sketch to locate solutions to an equation.
2013 STEP 2 Question 1.
2015 STEP 1 Question 1. Involves sketching a graph of the form $y=f(x)$ and then a graph of the form $y=f(x^2)$.
Making Waves from Nrich
Loch Ness from Nrich
You can find the STEP questions using the STEP database, where you can also search for more curve sketching questions. You can find more questions in the TBO booklet, on the Underground Mathematics website and in this document (which also has links to a couple of videos of curve sketching).
