To solve this problem, we need to consider the properties of a quadratic function whose vertex is also its y-intercept. Let's explore what must be true for such a quadratic function.
1. General Form of a Quadratic Function:
A quadratic function generally has the form [tex]\(f(x) = ax^2 + bx + c\)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are constants.
2. Vertex Form of a Quadratic Function:
The vertex form of a quadratic function is given by [tex]\(f(x) = a(x-h)^2 + k\)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
3. Vertex and y-Intercept:
For the vertex to be the same as the y-intercept, the vertex [tex]\((h, k)\)[/tex] must lie on the y-axis. This implies that the x-coordinate of the vertex, [tex]\(h\)[/tex], must be equal to 0. Consequently, the vertex must be [tex]\((0, k)\)[/tex].
4. Axis of Symmetry:
The axis of symmetry of a quadratic function is a vertical line that passes through the vertex. Given that the vertex is at [tex]\((0, k)\)[/tex], the axis of symmetry is the line [tex]\(x = 0\)[/tex].
5. Conclusion:
Since the vertex is on the y-axis, the axis of symmetry must be [tex]\(x = 0\)[/tex].
The correct statement is:
- The axis of symmetry for the function is [tex]\(x = 0\)[/tex].
Therefore, the answer is that the axis of symmetry for the function is [tex]\(x=0\)[/tex].
