Question

if the vertex of a quadratic functions is on the x-axis,what must be true about the number of x-intercepts? how do you know?

Answer 1

If the vertex of a quadratic function is on the x-axis, this means that the function has exactly one x-intercept.

This can be explained through the properties of a quadratic function, which can be expressed in the standard form \(y = ax^2 + bx + c\). The vertex form of a quadratic function is given by \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola.

1. **Vertex on the X-Axis**: If the vertex is on the x-axis, this means that the y-coordinate \(k\) of the vertex is 0. Therefore, the vertex is at the point \((h, 0)\).

2. **Nature of the Parabola**: The shape of the parabola (opening upwards or downwards) depends on the coefficient \(a\):
- If \(a > 0\), the parabola opens upwards.
- If \(a < 0\), the parabola opens downwards.

3. **One X-Intercept**: Since the vertex is the lowest point of the parabola (for \(a > 0\)) or the highest point (for \(a < 0\)), and it lies on the x-axis, the parabola must touch the x-axis at that vertex point but does not cross it. Therefore, in both cases (opening up or down), there is precisely one point where the function equals zero—this is the definition of having one x-intercept.

In conclusion, if the vertex of the quadratic function is on the x-axis, the function has exactly one x-intercept.