Question

The functions f(x) = x2 – 2 and g(x) = –x2 + 5 are shown on the graph.Explain how to modify the graphs of f(x) and g(x) to graph the solution set to the following system of inequalities. How can the solution set be identified? y > x2 – 2 y ≥ –x2 + 5

Answer 1

Two rigid transformations must be applied:

1. Vertical translation 7 units in the +y direction.
2. Reflection around the line y = 5.

The solution set can be identified by the fact that domain of quadratic functions is the real domain.

How to use rigid transformations

In the field of the Euclidean geometry rigid transformations are transformations applied on geometric loci such that Euclidean distances within the latter are conserved and herein we must determine what rigid transformations we should apply on f(x) to obtain g(x).

After a careful examination, we conclude that following two rigid transformations must be applied:

1. Vertical translation 7 units in the +y direction.
2. Reflection around the line y = 5.

The solution set represents the set of x-values such that the function exists. As both functions are quadratic equations, then their solutions are described the entire real domain.

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