What is the vertex of the function [tex]f(x)=x^2+12 x[/tex]?
A. [tex](6,-36)[/tex]
B. [tex](-6,0)[/tex]
C. [tex](6,0)[/tex]
D. [tex](6,-36)[/tex]
Answer (1)
Identify the coefficients: a = 1 and b = 12 .
Calculate the x-coordinate of the vertex: h = − 2 a b = − 6 .
Calculate the y-coordinate of the vertex: k = f ( − 6 ) = ( − 6 ) 2 + 12 ( − 6 ) = − 36 .
State the vertex: The vertex is ( − 6 , − 36 ) .
Explanation
Understanding the Problem We are given the quadratic function f ( x ) = x 2 + 12 x and asked to find its vertex. The vertex form of a quadratic function is f ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola. We can find the vertex by completing the square or by using the formula h = − 2 a b and k = f ( h ) , where a and b are the coefficients of the quadratic and linear terms, respectively.
Finding the x-coordinate of the vertex In our case, the quadratic function is f ( x ) = x 2 + 12 x , so a = 1 and b = 12 . We can find the x-coordinate of the vertex, h , using the formula: h = − 2 a b = − 2 ( 1 ) 12 = − 2 12 = − 6
Finding the y-coordinate of the vertex Now we can find the y-coordinate of the vertex, k , by plugging h = − 6 into the function: k = f ( − 6 ) = ( − 6 ) 2 + 12 ( − 6 ) = 36 − 72 = − 36
Stating the vertex Therefore, the vertex of the quadratic function f ( x ) = x 2 + 12 x is ( − 6 , − 36 ) .
Examples
Understanding the vertex of a parabola 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 you're modeling profit as a quadratic function, the vertex indicates the point of maximum profit. Knowing how to find the vertex allows you to optimize processes and make informed decisions in fields ranging from physics to economics.
