Which function has a vertex at the origin?
[tex]f(x)=(x+4)^2[/tex]
[tex]f(x)=(x-4)(x+4)[/tex]
[tex]f(x)=x(x-4)[/tex]
[tex]f(x)=-x^2[/tex]
Answer (2)
The vertex of f ( x ) = ( x + 4 ) 2 is ( − 4 , 0 ) .
The vertex of f ( x ) = ( x − 4 ) ( x + 4 ) is ( 0 , − 16 ) .
The vertex of f ( x ) = x ( x − 4 ) is ( 2 , − 4 ) .
The vertex of f ( x ) = − x 2 is ( 0 , 0 ) .
The function with a vertex at the origin is f ( x ) = − x 2 .
Explanation
Problem Analysis We are given four functions and we need to find the one with its vertex at the origin (0, 0). Let's analyze each function.
Analyzing f(x) = (x+4)^2
f ( x ) = ( x + 4 ) 2 : This is a quadratic function in vertex form, f ( x ) = a ( x − h ) 2 + k , where the vertex is at ( h , k ) . In this case, h = − 4 and k = 0 , so the vertex is at ( − 4 , 0 ) .
Analyzing f(x) = (x-4)(x+4)
f ( x ) = ( x − 4 ) ( x + 4 ) : This can be simplified to f ( x ) = x 2 − 16 . This is also a quadratic function, and its vertex is at ( 0 , − 16 ) .
Analyzing f(x) = x(x-4)
f ( x ) = x ( x − 4 ) : This can be expanded to f ( x ) = x 2 − 4 x . To find the vertex, we can use the formula x = − b / ( 2 a ) , where a = 1 and b = − 4 . So, x = − ( − 4 ) / ( 2 ∗ 1 ) = 2 . Then, f ( 2 ) = 2 ( 2 − 4 ) = − 4 . Thus, the vertex is at ( 2 , − 4 ) .
Analyzing f(x) = -x^2
f ( x ) = − x 2 : This is a quadratic function with a vertex at ( 0 , 0 ) because when x = 0 , f ( 0 ) = − 0 2 = 0 .
Conclusion Therefore, the function with a vertex at the origin is f ( x ) = − x 2 .
Examples
Understanding the vertex of a parabola is crucial in many real-world applications. For example, engineers use this concept to design parabolic mirrors and satellite dishes, where the vertex represents the optimal focal point. Similarly, in physics, projectile motion follows a parabolic path, and knowing the vertex helps determine the maximum height reached by the projectile. This knowledge is also useful in economics for modeling cost and revenue curves to find the point of maximum profit or minimum cost.
The function that has a vertex at the origin (0, 0) is f ( x ) = − x 2 . This is determined by analyzing the vertex of each given function. All other functions have vertices located at points other than the origin.
;
