### Why do this problem?

This challenge could be used as a lesson starter to check students'
ability to match curves with their equations. Finding other
functions which do not intersect with the existing curves except at
the endpoints can help students to develop a stronger sense of how
to manipulate functions and the corresponding effect on the
graph.

### Possible approach

This problem lends itself to discussion as a class or in small
groups. Give students a short time to consider how to identify
which is the $x$ and which is the $y$ axis, then share
reasoning.

Are there any of the curves they can identify straight away?
The curves could be matched using some numerical work to identify
key points on the graphs. Alternatively, students could consider
how they could sketch the less familiar curves by transforming the
functions which are more recognisable.

Considering the relationship between the functions which were
chosen for the problem may lead to insight into how to create more
curves with the same endpoints which do not intersect at any other
point in the region. Students could use graph-plotting software to
sketch these graphs and explain how they know there are no other
points of intersection.

### Key questions

Are there any curves which can be identified straight
away?

Which curves have similar shapes to each other? How can they
be distinguished?

Is there a relationship between any of the curves?

### Possible extension

Come up with ways of generating curves between any pair on the
grid, and prove that there are no intersections except at the end
points.

### Possible support

Work numerically to plot a few key points for each given function,
and use these to identify the $x$ and $y$ axes and each
curve.