The graph of [tex]f(x)=(x+3)^2+4[/tex] has its vertex at (3,4).
Choose the correct answer below.
True
False
Answer (2)
The vertex form of a quadratic function is f ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex.
Rewrite the given function f ( x ) = ( x + 3 ) 2 + 4 in vertex form: f ( x ) = ( x − ( − 3 ) ) 2 + 4 .
Identify the vertex as ( − 3 , 4 ) .
Compare the calculated vertex ( − 3 , 4 ) with the stated vertex ( 3 , 4 ) and conclude that the statement is false. The final answer is F a l se .
Explanation
Analyze the problem The given function is f ( x ) = ( x + 3 ) 2 + 4 . We need to determine if the vertex of the graph of this function is at ( 3 , 4 ) .
Recall 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.
Rewrite the function in vertex form We can rewrite the given function f ( x ) = ( x + 3 ) 2 + 4 in the vertex form as f ( x ) = 1 ( x − ( − 3 ) ) 2 + 4 . Comparing this with the general vertex form f ( x ) = a ( x − h ) 2 + k , we can identify the vertex as ( h , k ) = ( − 3 , 4 ) .
Compare the vertices The problem states that the vertex is at ( 3 , 4 ) . However, we found that the vertex is at ( − 3 , 4 ) . Since these two points are not the same, the statement is false.
Final Answer Therefore, the statement "The graph of f ( x ) = ( x + 3 ) 2 + 4 has its vertex at ( 3 , 4 ) " is false.
Examples
Understanding the vertex form of a quadratic equation is crucial in various real-world applications. For instance, when designing a parabolic reflector for a flashlight or satellite dish, knowing the vertex helps optimize the focus of light or signal. Similarly, in projectile motion, the vertex represents the maximum height reached by the projectile. By correctly identifying the vertex, engineers and scientists can accurately predict and control the behavior of these systems, ensuring efficient and precise outcomes. For example, if you want to model the trajectory of a ball thrown into the air, the function f ( x ) = − ( x − 2 ) 2 + 5 tells you that the maximum height of 5 meters is reached at a horizontal distance of 2 meters.
The statement that the graph of f ( x ) = ( x + 3 ) 2 + 4 has its vertex at ( 3 , 4 ) is false. The actual vertex of the function is ( − 3 , 4 ) . Therefore, the correct answer is False.
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