What is the vertex of the quadratic function [tex]f(x)=(x-8)(x-2)[/tex]?

Expand the quadratic function: f ( x ) = ( x − 8 ) ( x − 2 ) = x 2 − 10 x + 16 .
Find the x-coordinate of the vertex: x v ​ = − b / ( 2 a ) = 5 .
Substitute x v ​ into the function to find the y-coordinate: y v ​ = f ( 5 ) = − 9 .
The vertex of the quadratic function is ( 5 , − 9 ) ​ .

Explanation

Understanding the Problem We are given the quadratic function f ( x ) = ( x − 8 ) ( x − 2 ) and we want to find its vertex. The vertex of a quadratic function is the point where the function reaches its minimum or maximum value. To find the vertex, we can first expand the quadratic function into the standard form f ( x ) = a x 2 + b x + c , and then use the formula for the x-coordinate of the vertex, x v ​ = − b / ( 2 a ) . Finally, we can find the y-coordinate of the vertex by substituting x v ​ into the function, y v ​ = f ( x v ​ ) .

Expanding the Quadratic Function First, let's expand the quadratic function:


f ( x ) = ( x − 8 ) ( x − 2 ) = x 2 − 2 x − 8 x + 16 = x 2 − 10 x + 16
So, we have a = 1 , b = − 10 , and c = 16 .

Finding the x-coordinate of the Vertex Now, let's find the x-coordinate of the vertex using the formula x v ​ = − b / ( 2 a ) :

x v ​ = − ( − 10 ) / ( 2 ∗ 1 ) = 10/2 = 5

Finding the y-coordinate of the Vertex Next, we substitute x v ​ = 5 into the function f ( x ) to find the y-coordinate of the vertex:

y v ​ = f ( 5 ) = ( 5 − 8 ) ( 5 − 2 ) = ( − 3 ) ( 3 ) = − 9

Stating the Vertex Therefore, the vertex of the quadratic function is ( 5 , − 9 ) .

Examples
Understanding the vertex of a quadratic function 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 parabolic reflector, the vertex is the point where all incoming rays are focused. In business, the vertex can represent the point of maximum profit or minimum cost in a cost-benefit analysis. Knowing how to find the vertex allows you to optimize various processes and designs.

Answered by GinnyAnswer | 2025-07-04