## Problem

None of the following sketches is an accurate graph of the function $f(x)=\dfrac{x-5}{x^2-2x-3}.$

But which of these could be a sketch graph of $f(x)$?

• Try to explain why some of the sketches can’t represent this function.
• Which sketch graph do you think is the best representation of this function?

Once you have decided which graph(s) could represent this function, try to label key features, such as points where the graph crosses the axes. You could also try to sketch a graph that you think is a bit more true to the function than these graphs.

The sketch graphs are available separately as cards.

You may never have sketched a graph like this before, but we are asking you to think about what information the equation of the function gives you. It can be easier to think carefully about a function that you’re not familiar with.

Remember that a sketch graph needs to be as true to the function as necessary and this may depend on how we want to use the graph. It needs to show crucial features, but we don’t need to worry about exact distances and values where we don’t need them.