The vertex form of a quadratic function is [tex]$f(x)=a(x-h)^2+k$[/tex]. What is the vertex of each function? Match the function rule with the coordinates of its vertex.
(5,6)
(5,-9)
(-5,-6)
(-9,-5)
(6,9)
[tex]$f(x)=9(x+5)^2-6$[/tex]
[tex]$f(x)=5(x-6)^2+9$[/tex]
[tex]$f(x)=6(x+9)^2-5$[/tex]
[tex]$f(x)=9(x-5)^2+6$[/tex]
[tex]$f(x)=6(x-5)^2-9$[/tex]
Answer (1)
Identify the vertex form of a quadratic function: f ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex.
Extract h and k from each given function.
Determine the vertex coordinates ( h , k ) for each function.
Match each function with its corresponding vertex coordinates: ( − 5 , − 6 ) , ( 6 , 9 ) , ( − 9 , − 5 ) , ( 5 , 6 ) , ( 5 , − 9 ) .
Explanation
Understanding the Vertex Form The vertex form of a quadratic function is given by f ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola. We need to identify the vertex for each given function and match it with the provided coordinates.
Finding the Vertex of the First Function For the function f ( x ) = 9 ( x + 5 ) 2 − 6 , we can rewrite it as f ( x ) = 9 ( x − ( − 5 ) ) 2 + ( − 6 ) . Thus, h = − 5 and k = − 6 , so the vertex is ( − 5 , − 6 ) .
Finding the Vertex of the Second Function For the function f ( x ) = 5 ( x − 6 ) 2 + 9 , we have h = 6 and k = 9 , so the vertex is ( 6 , 9 ) .
Finding the Vertex of the Third Function For the function f ( x ) = 6 ( x + 9 ) 2 − 5 , we can rewrite it as f ( x ) = 6 ( x − ( − 9 ) ) 2 + ( − 5 ) . Thus, h = − 9 and k = − 5 , so the vertex is ( − 9 , − 5 ) .
Finding the Vertex of the Fourth Function For the function f ( x ) = 9 ( x − 5 ) 2 + 6 , we have h = 5 and k = 6 , so the vertex is ( 5 , 6 ) .
Finding the Vertex of the Fifth Function For the function f ( x ) = 6 ( x − 5 ) 2 − 9 , we have h = 5 and k = − 9 , so the vertex is ( 5 , − 9 ) .
Matching the Functions with Their Vertices Therefore, the matching is as follows:
f ( x ) = 9 ( x + 5 ) 2 − 6 matches with ( − 5 , − 6 ) .
f ( x ) = 5 ( x − 6 ) 2 + 9 matches with ( 6 , 9 ) .
f ( x ) = 6 ( x + 9 ) 2 − 5 matches with ( − 9 , − 5 ) .
f ( x ) = 9 ( x − 5 ) 2 + 6 matches with ( 5 , 6 ) .
f ( x ) = 6 ( x − 5 ) 2 − 9 matches with ( 5 , − 9 ) .
Examples
Understanding the vertex form of a quadratic equation is crucial in various real-world applications. For instance, when designing a parabolic mirror for a solar oven, knowing the vertex helps in focusing sunlight efficiently. Similarly, in sports, understanding the trajectory of a ball (which can be modeled as a parabola) requires identifying the vertex to determine the maximum height and range. This knowledge is also valuable in architecture, where parabolic arches are used for their structural strength and aesthetic appeal. By understanding the vertex form, we can optimize designs and predict outcomes in these diverse fields.
