Given that the parabola has a "curved v shape" and a "low point" at the point 5, we can infer that the vertex is at the point (5, k) for some value of k. This means the parabola opens upwards since it has a low point.
Let's analyze each statement:
A: The y-intercept and the x-intercepts share the same point.
- This is not generally true because the y-intercept is determined by the value of the parabola when x=0, and x-intercepts occur where the parabola intersects the x-axis (y=0). It is quite rare for these two to be the same point, especially since the vertex is at (5, k).
B: There is only one x-intercept.
- A parabola can have either 0, 1, or 2 x-intercepts. If the vertex is above the x-axis (k > 0), it has no x-intercepts. If it is on the x-axis (k = 0), it has 1 x-intercept. If it is below the x-axis (k < 0), it has 2 x-intercepts. Therefore, we cannot claim this statement is true without further information about k.
C: The y-intercept and the vertex share the same point.
- The vertex is at (5, k) and the y-intercept is at (0, f(0)), which generally will differ unless the specific form of the quadratic is such that these points coincide.
D: The vertex and the x-intercepts share the same point.
- Since the vertex is at (5, k), for the vertex and x-intercepts to coincide, the parabola would need to be tangent to the x-axis at the vertex (which occurs only if k = 0). Since there is no given information indicating this must be the case, we cannot confirm this statement is valid either.
Based on the information given, the best conclusion is that statement B is correct if k = 0 (it indicates one x-intercept), but in the general case, additional information on the parabola's equation is necessary to fully determine which of these statements is definitively true.
If we assume that the low point does refer strictly to one x-intercept occurrence at (5,0), then B: There is only one x-intercept can be considered true. Otherwise, the correct interpretation may vary based on the value of k.
1 answer