The parent function of the function [tex]g(x)=(x-h)^2+k[/tex] is [tex]f(x)=x^2[/tex]. The vertex of the function [tex]g(x)[/tex] is located at [tex](9,-8)[/tex]. What are the values of [tex]h[/tex] and [tex]k[/tex]?
[tex]g(x)=(x-[/tex]$\square$ [tex])^2+[/tex]$\square$
Answer (2)
The vertex form of a quadratic function is g ( x ) = ( x − h ) 2 + k , where ( h , k ) is the vertex.
Given the vertex ( 9 , − 8 ) , we directly identify h and k .
Therefore, h = 9 and k = − 8 .
The values are h = 9 and k = − 8 , so the answer is h = 9 , k = − 8 .
Explanation
Understanding the Problem The problem provides the vertex form of a quadratic function g ( x ) = ( x − h ) 2 + k and tells us that the vertex of the function is located at ( 9 , − 8 ) . We need to find the values of h and k .
Recalling Vertex Form The vertex form of a quadratic function is given by g ( x ) = ( x − h ) 2 + k , where ( h , k ) represents the coordinates of the vertex of the parabola.
Identifying h and k We are given that the vertex is ( 9 , − 8 ) . By comparing this with the vertex form ( h , k ) , we can directly identify the values of h and k . Therefore, h = 9 and k = − 8 .
Final Answer Thus, the values of h and k are 9 and − 8 , respectively.
Examples
Understanding vertex form is useful in various real-world applications. For example, engineers use quadratic functions to model the trajectory of a projectile, such as a ball thrown in the air. The vertex form helps them quickly determine the maximum height the ball will reach and the horizontal distance at which it occurs. Similarly, architects use quadratic functions to design arches and bridges, where the vertex represents the highest or lowest point of the structure. By knowing the vertex, they can ensure the structural integrity and aesthetic appeal of their designs. Understanding vertex form also helps in optimizing various processes, such as maximizing profit or minimizing cost in business scenarios.
The vertex of the function g ( x ) = ( x − h ) 2 + k is located at (9, -8), which means that h = 9 and k = − 8 . Therefore, the answer is h = 9 and k = − 8 .
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