Use the vertex formula to determine the vertex of the graph of the function. f(x) = -x² + 8x + 1 (x, y) = ( ) Write the function in standard form. f(x) =

To determine the vertex of the graph of the function f ( x ) = − x 2 + 8 x + 1 using the vertex formula, we need to follow these steps:

Identify the coefficients : The function is in the form f ( x ) = a x 2 + b x + c where a = − 1 , b = 8 , and c = 1 .

Use the vertex formula : The x-coordinate of the vertex h is given by the formula h = 2 a − b ​ .
h = 2 ( − 1 ) − 8 ​ = − 2 − 8 ​ = 4

Substitute x = 4 to find the y-coordinate :
Plug x = 4 back into the function to find k , the y-coordinate.
k = f ( 4 ) = − 4 2 + 8 ( 4 ) + 1


k = − 16 + 32 + 1 k = 17
Therefore, the vertex of the function is ( 4 , 17 ) .
Next, let's write the function in standard form. The standard form of a quadratic function is f ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex.

The vertex is ( 4 , 17 ) , and a = − 1 , so:

f ( x ) = − ( x − 4 ) 2 + 17
Thus, the vertex is ( 4 , 17 ) , and the function in standard form is f ( x ) = − ( x − 4 ) 2 + 17 .

Answered by AvaCharlotteMiller | 2025-07-21