What is the $y$-value of the vertex of the function $f(x)=-(x-3)(x+11)$?

Expand the function: f ( x ) = − x 2 − 8 x + 33 .
Find the x -coordinate of the vertex: x v ​ = − 2 a b ​ = − 4 .
Substitute x v ​ into the function to find the y -coordinate: f ( − 4 ) = 49 .
The y -value of the vertex is 49 ​ .

Explanation

Expand the function First, we need to find the x -coordinate of the vertex of the function f ( x ) = − ( x − 3 ) ( x + 11 ) . To do this, we first expand the function:

Expanding Expanding the function, we get: f ( x ) = − ( x 2 + 11 x − 3 x − 33 ) = − ( x 2 + 8 x − 33 ) = − x 2 − 8 x + 33

Find x-coordinate of vertex Now, we can find the x -coordinate of the vertex using the formula x v ​ = − 2 a b ​ , where a = − 1 and b = − 8 :
x v ​ = − 2 ( − 1 ) − 8 ​ = − − 2 − 8 ​ = − 4

Find y-coordinate of vertex Next, we substitute x v ​ = − 4 into the function f ( x ) to find the y -coordinate of the vertex: f ( − 4 ) = − ( − 4 ) 2 − 8 ( − 4 ) + 33 = − 16 + 32 + 33 = 16 + 33 = 49

Final Answer Therefore, the y -value of the vertex is 49.


Examples
Understanding the vertex of a parabola is useful in many real-world applications. For example, if you are throwing a ball, the path of the ball can be modeled by a parabola. The vertex of the parabola represents the highest point the ball will reach. Similarly, in business, if you are trying to maximize profit, the profit function can sometimes be modeled by a parabola, and the vertex will represent the point of maximum profit. Knowing how to find the vertex allows you to solve these types of optimization problems.

Answered by GinnyAnswer | 2025-07-04

The y -value of the vertex of the function f ( x ) = − ( x − 3 ) ( x + 11 ) is 49, as calculated by expanding the function and identifying the vertex coordinates.
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Answered by Anonymous | 2025-07-06