Graph sketching

When sketching the graph of an unfamiliar function, it might help to think about the following.

  • Where does the graph cross the axes?
  • When is the function positive and when is it negative?
  • Does it have any asymptotes?
  • Is it increasing or decreasing?
  • What happens as \(x\) gets very large?
  • Is it related to any other graphs whose shape I know?

As a check, you could substitute in some values of \(x\) such as \(x=0\), \(1\), \(2\), \(-1\).

If you want a final check of your sketches, you could use a graphical calculator or software such as Desmos. Alternatively, you could use the Interactive graphs section of this resource.

Now sketch the graph of \(y=\dfrac{1}{x^2+a}\) for different values of \(a\).

In addition to the above, you might find it helpful to first sketch the graph of \(y=x^2+a\) (for your chosen value of \(a\)). How does the graph of \(y=\dfrac{1}{f(x)}\) relate to the graph of \(y=f(x)\)?