Question

How do you find the max or minimum of `f(x)=-20x+5x^2+9`?

Answer 1

See below.

Explanation:

Because the function is `f(x)=5x^2-20x+9`, and the coefficient of the `x^2` term is positive, it will achieve a minimum at its vertex. To find the minimum, we need to express the parabola in vertex form.

`f(x)=5x^2-20x+9`
`f(x)=5(x^2-4x)+9`
To complete the square in the parenthesis, we need to add a `4` into the expression, but because there is a `5` on the outside, we actually add `20`. Now, because we added a `20` into the function, we need to subtract a `20` as well.

`f(x)=5(x^2-4x+4)+9-20`
`f(x)=5(x-2)^2-11`
Thus, the vertex is `(2, -11)`, and is the minimum.

graph{5x^2-20x+9 [-40, 40, -20.84, 20.84]}