Which must be true of a quadratic function whose vertex is the same as its $y$-intercept?

A. The axis of symmetry for the function is $x=0$.
B. The axis of symmetry for the function is $y=0$.
C. The function has no $x$-intercepts.
D. The function has $1 x$-intercept.

The quadratic function is of the form f ( x ) = a x 2 + c .
The axis of symmetry is x = − 2 a b ​ = 0 .
The axis of symmetry for the function is x = 0 must be true.
Therefore, the answer is The axis of symmetry for the function is x = 0 ​ .

Explanation

Understanding the Problem We are given a quadratic function whose vertex is the same as its y -intercept. We need to determine which of the given statements must be true.

Finding the Quadratic Form Let the quadratic function be f ( x ) = a x 2 + b x + c . The y -intercept is the point where x = 0 , so the y -intercept is ( 0 , c ) . The vertex of the quadratic function is given by x = − 2 a b ​ . Since the vertex is the same as the y -intercept, the x -coordinate of the vertex must be 0. Therefore, − 2 a b ​ = 0 , which implies b = 0 . Thus, the quadratic function is of the form f ( x ) = a x 2 + c .

Determining the Axis of Symmetry The axis of symmetry is given by x = − 2 a b ​ . Since b = 0 , the axis of symmetry is x = 0 . If a and c have the same sign, then the function has no x -intercepts. If a and c have opposite signs, then the function has two x -intercepts. If c = 0 , then the function has one x -intercept at x = 0 . Therefore, the axis of symmetry for the function is x = 0 must be true.

Conclusion The axis of symmetry for the function is x = 0 . This means the quadratic is symmetric about the y-axis. The other options are not necessarily true. For example, f ( x ) = x 2 + 1 has no x-intercepts, while f ( x ) = x 2 − 1 has two x-intercepts. Also, f ( x ) = x 2 has one x-intercept.


Examples
Understanding quadratic functions is crucial in various fields, such as physics, engineering, and economics. For example, the trajectory of a projectile under gravity can be modeled by a quadratic function. The vertex of the parabola represents the maximum height reached by the projectile. Similarly, in economics, quadratic functions can be used to model cost and revenue curves, where the vertex represents the point of maximum profit or minimum cost. By analyzing the properties of quadratic functions, we can make informed decisions and optimize outcomes in real-world scenarios.

Answered by GinnyAnswer | 2025-07-05

For a quadratic function where the vertex equals its y -intercept, the axis of symmetry must be x = 0 . This is due to the fact that the vertex's x -coordinate is derived from the parameter b being zero. Therefore, the answer is option A: The axis of symmetry for the function is x = 0 .
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Answered by Anonymous | 2025-07-12