Let's examine the properties of a quadratic function whose vertex is the same as its [tex]\(y\)[/tex]-intercept.
1. Axis of Symmetry:
A quadratic function of the form [tex]\(y = ax^2 + bx + c\)[/tex] generally has its vertex at the point [tex]\(\left(\frac{-b}{2a}, f\left(\frac{-b}{2a}\right)\right)\)[/tex]. However, if the vertex is the same as the [tex]\(y\)[/tex]-intercept, it means the vertex lies on the [tex]\(y\)[/tex]-axis. The [tex]\(y\)[/tex]-intercept occurs when [tex]\(x = 0\)[/tex]. Therefore, the vertex must be at the point [tex]\((0, c)\)[/tex].
For the vertex to be at [tex]\((0, c)\)[/tex], the quadratic function must have no linear term, [tex]\(bx\)[/tex], because any non-zero [tex]\(b\)[/tex] would shift the vertex horizontally. Thus, the quadratic function should be of the form [tex]\(y = ax^2 + c\)[/tex].
For such a function, the axis of symmetry is a vertical line that passes through the vertex. Since the vertex is at [tex]\((0, c)\)[/tex], the axis of symmetry must be at [tex]\(x = 0\)[/tex].
2. Number of [tex]\(x\)[/tex]-intercepts:
The [tex]\(x\)[/tex]-intercepts of a quadratic function are the points where the function intersects the [tex]\(x\)[/tex]-axis. These points are found by solving the equation [tex]\(ax^2 + c = 0\)[/tex].
- If [tex]\(a\)[/tex] and [tex]\(c\)[/tex] are both non-zero, and there is a single solution to [tex]\(ax^2 + c = 0\)[/tex], it means the vertex [tex]\((0, c)\)[/tex] must be the only point where the function touches the [tex]\(x\)[/tex]-axis. When the vertex is the same as the [tex]\(y\)[/tex]-intercept, and if the function does intersect the [tex]\(x\)[/tex]-axis, it will only touch it at one point.
The quadratic function [tex]\(y = ax^2 + c\)[/tex] will have a single [tex]\(x\)[/tex]-intercept if it touches the axis precisely at one point, which in this case will be when [tex]\(a\)[/tex] and [tex]\(c\)[/tex] ensure that the vertex lies on the [tex]\(x\)[/tex]-axis at [tex]\((0, 0)\)[/tex].
Based on these findings, we can conclude the following statements:
- The axis of symmetry for the function is [tex]\(x = 0\)[/tex].
- The function has one [tex]\(x\)[/tex]-intercept.
Thus, the correct statements must be:
- The axis of symmetry for the function is [tex]\(x=0\)[/tex].
- The function has [tex]\(1\)[/tex] [tex]\(x\)[/tex]-intercept.
