Question

How do you find the range of a quadratic equation `f(x) = -x^2 + 14x - 48`?

Answer 1

Given `f(x) = -x^2+14x-48`

`f'(x) = -2x+14`
for critical point(s)
`f'(x)= 0`
`-2x+14 = 0`
`x=7` for the only critical point

We can either use our knowledge of quadratics or take the second derivative (`f''(x)=-2 rarr` slope is always decreasing) to observe that this point is a maximum.

Therefore `f(x)` has a maximum when `x=7`
`f(7) = -(49) + 98 -48 = 1`

So the range of `f(x)` is `[-oo,+1]`