What is the vertex of the quadratic function $f(x)=(x-8)(x-2)$?
( $\square$ , $\square$ )
Answer (1)
Expand the quadratic function: f ( x ) = x 2 − 10 x + 16 .
Find the x-coordinate of the vertex: x v = − 2 a b = 5 .
Find the y-coordinate of the vertex: y v = f ( 5 ) = − 9 .
State the vertex: ( 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.
Expanding the Quadratic Function First, we need to expand the quadratic function into the standard form f ( x ) = a x 2 + b x + c . Expanding ( x − 8 ) ( x − 2 ) , we get:
f ( x ) = x 2 − 2 x − 8 x + 16 = x 2 − 10 x + 16
So, a = 1 , b = − 10 , and c = 16 .
Finding the x-coordinate of the Vertex The x-coordinate of the vertex, x v , can be found using the formula x v = − 2 a b . In our case, a = 1 and b = − 10 , so:
x v = − 2 ( 1 ) − 10 = 2 10 = 5
Finding the y-coordinate of the Vertex Now, we need to find the y-coordinate of the vertex, y v , by substituting x v = 5 into the function f ( x ) = x 2 − 10 x + 16 :
y v = f ( 5 ) = ( 5 ) 2 − 10 ( 5 ) + 16 = 25 − 50 + 16 = − 9
Stating the Vertex Therefore, the vertex of the quadratic function is ( 5 , − 9 ) .
Examples
Understanding the vertex of a quadratic function is very useful in many real-world applications. For example, if you are throwing a ball, the path of the ball can be modeled by a quadratic function. The vertex of this function represents the highest point the ball will reach. Similarly, businesses can use quadratic functions to model their profit, where the vertex represents the point of maximum profit. Knowing how to find the vertex allows you to determine these maximum or minimum values in various scenarios.
