This task was inspired by an article in the ATM journal “Mathematics Teaching”, issue 116 by Tasos Patronis titled “Exploring Exponentials”.
He makes the point that many pupils do not have a sense of the types of phenomena that can be captured by mathematical functions. I feel this task helps to illustrate the key property of exponential growth and how this differs from linear growth.
The exponential graph arises as a matter of course from this. Pupils can be asked to speculate on the values where x = -1 and x = -2. They could be asked to suggest an equation for this curve.
Some may feel this task is too scaffold-ed. It might be nice to offer the initial problem without the table of values and the graph as learners own ideas may prompt interesting dialogue.
Credit: @chrismcgrane84 and Tasos Patronis