What is the $y$-value of the vertex of the function $f(x)=-(x+8)(x-14)$?

Expand the function f ( x ) = − ( x + 8 ) ( x − 14 ) to get f ( x ) = − x 2 + 6 x + 112 .
Find the x -coordinate of the vertex using x v ​ = − 2 a b ​ , which gives x v ​ = 3 .
Substitute x v ​ = 3 into the function to find the y -coordinate of the vertex: f ( 3 ) = − 9 + 18 + 112 = 121 .
The y -value of the vertex is 121 ​ .

Explanation

Rewrite the function We are given the function f ( x ) = − ( x + 8 ) ( x − 14 ) and asked to find the y -value of the vertex. The vertex represents the maximum or minimum point of the parabola. To find it, we first need to rewrite the function in the standard quadratic form f ( x ) = a x 2 + b x + c .

Expand the function Expanding the given function, we have:


f ( x ) = − ( x + 8 ) ( x − 14 ) f ( x ) = − ( x 2 − 14 x + 8 x − 112 ) f ( x ) = − ( x 2 − 6 x − 112 ) f ( x ) = − x 2 + 6 x + 112
So, a = − 1 , b = 6 , and c = 112 .

Find the x-coordinate of the vertex The x -coordinate of the vertex is given by the formula x v ​ = − 2 a b ​ . Plugging in the values of a and b , we get:

x v ​ = − 2 ( − 1 ) 6 ​ = − − 2 6 ​ = 3

Find the y-coordinate of the vertex Now, we substitute the x -coordinate of the vertex, x v ​ = 3 , into the function f ( x ) to find the y -coordinate of the vertex:

f ( 3 ) = − ( 3 ) 2 + 6 ( 3 ) + 112 f ( 3 ) = − 9 + 18 + 112 f ( 3 ) = 9 + 112 f ( 3 ) = 121
Thus, the y -value of the vertex is 121.

Final Answer Therefore, the y -value of the vertex of the function f ( x ) = − ( x + 8 ) ( x − 14 ) is 121.

Examples
Understanding the vertex of a parabola is useful in many real-world applications. For example, if you are launching a projectile, the vertex represents the maximum height the projectile will reach. Similarly, if you are designing a suspension bridge, understanding the parabolic shape and its vertex helps in calculating the tension and load distribution. In business, if you model profit as a quadratic function, the vertex can represent the maximum profit you can achieve. These applications highlight the importance of understanding quadratic functions and their properties.

Answered by GinnyAnswer | 2025-07-04

The y -value of the vertex of the function f ( x ) = − ( x + 8 ) ( x − 14 ) is 121 , derived by expanding the function, finding the x -coordinate of the vertex, and substituting back into the function to find the y -coordinate.
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Answered by Anonymous | 2025-07-06