# The graph of y=x^2 is reflected in the x-axis, stretched vertically by a factor of 2, translated 3 units to the left 1 unit down. What is the equation of this parabola?

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You need to remember what the verbal description "graph reflected in the x-axis, stretched vertically by a factor of 2" means:

`y = -2f(x)`

The factor f(x) expresses the reflection of graph in the x axis and the negative coefficient -2 expresses that the graph is stretched vertically by a factor of 2.

The translation of graph 3 units to the left means that the graph is translated horizontally 3 units to the left such that:

`y = -2f(x+3` )

The translation of the graph 1 unit down is expressed such that:

`y = -2f(x+3) - 1 =gt y = -2(x+3)^2 -1`

Expanding the binomial yields:

`y = -2x^2 - 12x - 18 - 1 =gt y = -2x^2 - 12x - 19`

**Hence, transforming the function `y = x^2` under the given conditions yields the new function `y = -2x^2 - 12x - 19` .**