In the following exercise, find the coordinates of the vertex.
[tex]f(x)=3(x-1)^2+2[/tex]
The vertex is $\square$. (Type an ordered pair.)
Answer (2)
Recognize the vertex form of a parabola: f ( x ) = a ( x − h ) 2 + k , where the vertex is ( h , k ) .
Identify h and k from the given function f ( x ) = 3 ( x − 1 ) 2 + 2 .
Determine the vertex coordinates: ( h , k ) = ( 1 , 2 ) .
State the vertex as an ordered pair: ( 1 , 2 ) .
Explanation
Problem Analysis The given function is f ( x ) = 3 ( x − 1 ) 2 + 2 . We need to find the vertex of this function.
Vertex Form The function is already in vertex form, which is given by f ( x ) = a ( x − h ) 2 + k , where ( h , k ) represents the vertex of the parabola.
Identifying h and k Comparing the given function f ( x ) = 3 ( x − 1 ) 2 + 2 with the vertex form f ( x ) = a ( x − h ) 2 + k , we can identify the values of h and k . Here, a = 3 , h = 1 , and k = 2 .
Stating the Vertex Therefore, the vertex of the parabola is ( h , k ) = ( 1 , 2 ) .
Examples
Understanding the vertex of a parabola is crucial in various real-world applications. For instance, when designing a suspension bridge, engineers need to determine the lowest point of the cable's curve, which can be modeled as a parabola. Similarly, in projectile motion, the vertex represents the maximum height reached by an object, helping athletes and engineers optimize trajectories. Knowing how to find the vertex allows for efficient problem-solving in these scenarios.
The vertex of the function f ( x ) = 3 ( x − 1 ) 2 + 2 is found to be the ordered pair ( 1 , 2 ) . This conclusion is based on identifying parameters h and k from the vertex form of the parabola. Hence, the vertex is ( 1 , 2 ) .
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