Question

How is the graph of `h(x)=-7-x^2` related to the graph of `f(x)=x^2`?

Answer 1

Rotates the graph by 180 degrees bout its vertex and then lowers it by 7.

Explanation:

`-x^2` shape is `nn` whilst `x^2` is `uu`

The vertex for each being in the same place of `(x,y)->(0,0)`

The `-7` of `-7-x^2` lowers the graph of `-x^2` by 7

Example Consider `y=-x^2`

Set `x_1` as 2

Then `y_1=-(2^2)=-4` Call this point 1

`color(green)(P_1->(x_1,y_1)=(2,-4))`
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Vertically lower this point by 7 and call the new point `P_2`

So just considering the coordinates we have:

`color(blue)(P_2->(x_1,color(white)("d")y_1-7)=(2,-4-7)=(2,-11))`

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Compare to `y_2=(x_2)^2-7`

Set `x_2=x_1` then we have:

`color(blue)(y_2=-(x_1)^2-7 => -2^2-7 = -11 ->P_2)`