Write an equation for the parabola that has a vertex at (-2,3) and with a vertical stretch of 5. Please explain fully.
Start with the "standard" parabola `y=x^2.` First we can take care of the vertical scale factor we need. The parabola described by
accomplishes the vertical stretch. This still has vertex `(0,0)` though. To change that, replace `x` by `x+2` and `y` by `y-3.` This gives
`y-3=5(x+2)^2,` or equivalently `y=5(x+2)^2+3.` Since `5(x+2)^2>=0` for all `x,` the right hand side takes the minimum value of 3 when `x=-2,` proving that the vertex is `(-2,3).`
Thus the equation we want is