Which function has a vertex at $(2,-9)$?

A. $f(x)=-(x-3)^2$
B. $f(x)=(x+8)^2$
C. $f(x)=(x-5)(x+1)$
D. $f(x)=-(x-1)(x-5)$

Analyze each function to find its vertex.
The vertex of f ( x ) = − ( x − 3 ) 2 is (3, 0).
The vertex of f ( x ) = ( x + 8 ) 2 is (-8, 0).
The vertex of f ( x ) = ( x − 5 ) ( x + 1 ) is (2, -9).
The vertex of f ( x ) = − ( x − 1 ) ( x − 5 ) is (3, 4).
The function with a vertex at (2, -9) is f ( x ) = ( x − 5 ) ( x + 1 ) ​ .

Explanation

Problem Analysis We are given four functions and asked to identify the one with a vertex at (2, -9). The vertex form of a quadratic function is f ( x ) = a ( x − h ) 2 + k , where (h, k) is the vertex. We will analyze each function to find its vertex.

Analyzing the first function

f ( x ) = − ( x − 3 ) 2 . This is in vertex form with a = − 1 , h = 3 , and k = 0 . The vertex is at (3, 0), which is not (2, -9).

Analyzing the second function

f ( x ) = ( x + 8 ) 2 . This is in vertex form with a = 1 , h = − 8 , and k = 0 . The vertex is at (-8, 0), which is not (2, -9).

Analyzing the third function

f ( x ) = ( x − 5 ) ( x + 1 ) . To find the vertex, we first expand the expression: f ( x ) = x 2 − 5 x + x − 5 = x 2 − 4 x − 5 . The x-coordinate of the vertex is given by x = − b / ( 2 a ) , where a = 1 and b = − 4 . Thus, x = − ( − 4 ) / ( 2 ∗ 1 ) = 4/2 = 2 . The y-coordinate of the vertex is f ( 2 ) = ( 2 ) 2 − 4 ( 2 ) − 5 = 4 − 8 − 5 = − 9 . Therefore, the vertex is (2, -9).

Analyzing the fourth function

f ( x ) = − ( x − 1 ) ( x − 5 ) . Expanding this, we get f ( x ) = − ( x 2 − 5 x − x + 5 ) = − ( x 2 − 6 x + 5 ) = − x 2 + 6 x − 5 . The x-coordinate of the vertex is x = − b / ( 2 a ) , where a = − 1 and b = 6 . Thus, x = − 6/ ( 2 ∗ ( − 1 )) = − 6/ ( − 2 ) = 3 . The y-coordinate of the vertex is f ( 3 ) = − ( 3 ) 2 + 6 ( 3 ) − 5 = − 9 + 18 − 5 = 4 . Therefore, the vertex is (3, 4), which is not (2, -9).

Final Answer The function with a vertex at (2, -9) is f ( x ) = ( x − 5 ) ( x + 1 ) .


Examples
Understanding the vertex of a quadratic function is crucial in various real-world applications. For instance, when designing a parabolic mirror for a telescope, the vertex represents the point where light rays converge, ensuring optimal focus. Similarly, in projectile motion, the vertex indicates the maximum height reached by an object, aiding in trajectory calculations. By identifying the vertex, engineers and scientists can optimize designs and predict outcomes in fields ranging from optics to sports.

Answered by GinnyAnswer | 2025-07-04