What is the vertex of the quadratic function [tex]$f(x)=(x-6)(x+2)$[/tex]?
Answer (1)
Expand the quadratic function: f ( x ) = x 2 − 4 x − 12 .
Find the x-coordinate of the vertex: x v = − 2 a b = 2 .
Find the y-coordinate of the vertex: y v = f ( 2 ) = − 16 .
State the vertex: ( 2 , − 16 ) .
Explanation
Understanding the Problem We are given the quadratic function f ( x ) = ( x − 6 ) ( x + 2 ) and we need 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 expand the given quadratic function to the standard form f ( x ) = a x 2 + b x + c :
f ( x ) = ( x − 6 ) ( x + 2 ) = x 2 + 2 x − 6 x − 12 = x 2 − 4 x − 12 So, we have a = 1 , b = − 4 , and c = − 12 .
Finding the x-coordinate of the Vertex Next, we find the x-coordinate of the vertex using the formula x v = − 2 a b :
x v = − 2 ( 1 ) − 4 = 2 4 = 2 So, the x-coordinate of the vertex is 2 .
Finding the y-coordinate of the Vertex Now, we substitute x v = 2 into the function f ( x ) to find the y-coordinate of the vertex, y v = f ( x v ) :
y v = f ( 2 ) = ( 2 ) 2 − 4 ( 2 ) − 12 = 4 − 8 − 12 = − 16 So, the y-coordinate of the vertex is − 16 .
Stating the Vertex Therefore, the vertex of the quadratic function is ( 2 , − 16 ) .
Examples
Understanding the vertex of a quadratic function is crucial in various real-world applications. For instance, if you're launching a projectile, the vertex represents the maximum height the projectile will reach. Similarly, in business, if a quadratic function models the profit of a product, the vertex indicates the production level that maximizes profit. Knowing how to find the vertex allows you to optimize outcomes in these scenarios.
