Which function has a vertex at the origin?

A. [tex]f(x)=(x+4)^2[/tex]
B. [tex]f(x)=x(x-4)[/tex]
C. [tex]f(x)=(x-4)(x+4)[/tex]
D. [tex]f(x)=-x^2[/tex]

Analyze each function to find its vertex.
f ( x ) = ( x + 4 ) 2 has vertex ( − 4 , 0 ) .
f ( x ) = x ( x − 4 ) has vertex ( 2 , − 4 ) .
f ( x ) = ( x − 4 ) ( x + 4 ) has vertex ( 0 , − 16 ) .
f ( x ) = − x 2 has vertex ( 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 whose vertex is at the origin (0, 0). Let's analyze each function.

Finding Vertices

f ( x ) = ( x + 4 ) 2 : This is a parabola in vertex form f ( x ) = a ( x − h ) 2 + k , where the vertex is ( h , k ) . In this case, h = − 4 and k = 0 , so the vertex is ( − 4 , 0 ) .

f ( x ) = x ( x − 4 ) = x 2 − 4 x : To find the vertex, we can complete the square: f ( x ) = ( x 2 − 4 x + 4 ) − 4 = ( x − 2 ) 2 − 4 . The vertex is ( 2 , − 4 ) .

f ( x ) = ( x − 4 ) ( x + 4 ) = x 2 − 16 : This is also in vertex form f ( x ) = a ( x − h ) 2 + k , where h = 0 and k = − 16 . So the vertex is ( 0 , − 16 ) .

f ( x ) = − x 2 : This is in vertex form with h = 0 and k = 0 . So the vertex is ( 0 , 0 ) .

Identifying the Function The function f ( x ) = − x 2 has its vertex at the origin (0, 0).


Examples
Understanding the vertex of a parabola is crucial in various real-world applications. For instance, when designing a parabolic mirror for a telescope, the vertex represents the point where light rays converge, ensuring optimal focus. Similarly, in projectile motion, the vertex of the parabolic trajectory indicates the maximum height reached by the projectile. Knowing how to find the vertex allows engineers and scientists to optimize designs and predict outcomes in these scenarios.

Answered by GinnyAnswer | 2025-07-04

The function that has a vertex at the origin (0, 0) is Option D: f ( x ) = − x 2 . The vertex of this function is easily determined to be at (0, 0) because it is in the standard vertex form. The other options have vertices located at different coordinates.
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Answered by Anonymous | 2025-07-06