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We have learnt that the curvature of a function like $f(x)=x^2$ can be computed by its second derivative. However, a question came to my mind on the possible transformations which saves the shape of the graph, however the curvature is changed. For example, $-f(x)=-x^2$, changes the curvature, however its shape is not similar to $f(x)=x^2$.

Any suggestions on search keywords are welcome, too.

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Geometrically, all parabolas are similar, whether they open up or down; whether they are wide or narrow. This can be verified by taking the second derivative.

I'm not sure what you intended with your example, but if you divide both sides of $-f(x) = -x^2$, it appears identical to $f(x)=x^2$.

I think the only way to change the curvature of a parabola is to make it a cubic function (for example), but this is not a simple transformation.

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    Thanks for the answer. By the way, I have used the term function to denote any function, e.g. how to change the curvature of $\tan(x)$ or $x^3$ such that the curves are similar visually. This might not be possible, or it may be, however I do not know ... – 2017-02-07
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    If you restrict the domain of $tan(x)$, you should be able to apply transformations so it is similar *visually* to $x^3$. You could experiment at desmos.com (among many others). But they won't be the same function, i.e., $tan(ax+b)+c \neq x^3$ for all $x$. – 2017-02-07
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The curvature of a single variable real-valued function $f$ is given by $$k=\frac{|f''(x)|}{(1+(f'(x))^2)^{3/2}},$$ and this gives the same value for $f(x)=x^2$ and $f(x)=-x^2$, i.e. $$k=\frac{1}{(1+4x^2)^{3/2}}.$$

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    So, the transformation $-f(x)$ does not change the curvature. Are there any transformations which does the task? – 2017-02-07