Question

A parabola has a maxima at `(-4,7)` and passes through `(1,1)`. Which is the other point with same ordinate that it passes through?

Answer 1

Parabola also passes through `(-9,1)`

Explanation:

We can use here vertex form of equation for parabola. As it has a maximum at `(-4,7)`, it will be of the form

`y=-a(x+4)^2+7`

Note that at `(-4,7)`, `y` has a maximum value of `-7` as `a(x+4)^2=0` and at other places as it is negative, value of `y` is far less.

Now as `y=-a(x+4)^2+7` passes through `(1,1)`, we have

`1=-a(1+4)^2+7` or `1=-25a+7` i.e. `25a=6` and

`a=6/25` and equation of parabola is `y=-6/25(x+4)^2+7`

Note that this form of equation is symmetric w.r.t. `x+4=0` and hence The point symmetric to `(1,1)` will have abscissa given by `(x+1)/2=-4` i.e. `x=-9` and then

`y=-6/25(-9+4)^2+=-6/25xx25+7=1` i.e.

Parabola also passes through `(-9,1)`
graph{-6/25(x+4)^2+7 [-14.17, 5.83, -2.2, 7.8]}