Question

How do I use the vertex formula to determine the vertex of the graph for `f(x) = −x^2 + 18x − 75`?

Answer 1

The vertex formula has the form
`y=m(x-a)^2+b` for constants `m, a, b`
with a vertex at `(a,b)`

Converting `f(x)=-x^2+18x-75`
into the vertex formula form is a matter of "completing the square"

`(fx) = (-1)(x^2-18x+9^2) -75+9^2`

`= (-1)(x-9)^2+6`

The vertex is at `(9,6)`

Answer 2

The vertex formula says that the graph of `y=ax^2+bx+c` has its vertex at the point where `x = -b/(2a)`.

In this equation, we have `a=-1` and `b=18`.

The formula tells us that the `x`-coordinate of the vertex is:

`x=-(18)/(2(-1)) = -18/-2 =-(-9)=9`

To find the `y`-coordinate, use the equation and `x=9`:

`y=-(9)^2+18(9)-75`

`y=-81+ 162-75 = 162 - 156`

`y= 6`

The vertex is `(9,6)`