Please do not use a graphic calculator or graphing software in this resource!

Imagine you were asked to draw graphs of the following functions. Make a note of what you would think about in each case.

  • \(f(x)=3x-4\)

  • \(f(x)=x^2+7x+10\)

  • \(f(x)=\dfrac{1}{x}-1\)

  • \(f(x)=\dfrac{-1}{x^2-1}\)

Remember that a sketch graph needs to show the important features of a function and how the function behaves as the input values change. Sketching the graph of a function is a very different process from plotting the graph. It doesn’t need to be drawn to scale and we can’t draw any conclusions about distances in sketch graphs. This is one reason why it is important to label points of intersection with the axes and other features.

Did you think about the same things for each function? Did you think of them in the same order?

  • Do we know the rough shape of the graph? Is the graph a transformation of a graph we know?

  • Does the graph cross or meet the axes? If so, where?

  • Are there any values of \(x\) for which the function isn’t defined? If so, what happens when \(x\) is close to these values?

  • Does the graph have any asymptotes?

  • When is the value of the function positive? When is it negative?

  • What happens if \(x\) has a large positive value or a large negative value?