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A curve has a slope of $2e^{2x}$ for all values of $(x, y)$.

Determine which of the following equations could be the equation of the curve if it passes through the point $(0, -1)$.

$y={ e }^{ 2x }$

$y={ e }^{ 2x }-1$

$y={ e }^{ 2x }-2$

$y={ 2e }^{ 2x }-2$