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How Many Solutions?

Find all the solutions to the this equation.


Investigate the graphs of y = [1 + (x - t)^2][1 + (x + t^)2] as the parameter t varies.

Power Up

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

Curve Match

Age 16 to 18
Challenge Level


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.