To determine the truth of each statement regarding a parabola, let's analyze each one:
-
The y-intercept and the vertex share the same point.
This statement is not always true. The y-intercept is found where \( x = 0 \), and the vertex can be at a different location. Thus, this statement is false. -
There is only one x-intercept.
This is true for parabolas that touch the x-axis at exactly one point (these are called "perfect squares"), but not for all parabolas. Most parabolas will have either two x-intercepts or none. Therefore, this statement is not universally true. -
The y-intercept and the x-intercepts share the same point.
This statement is generally false. The y-intercept occurs at \( y \) when \( x = 0 \), while x-intercepts occur where \( y = 0 \). They do not necessarily coincide. -
The vertex and the x-intercepts share the same point.
This is also generally false. The vertex can be above or below the x-axis, and even if it is below or at the x-axis, it can still be distinct from the x-intercepts.
Based on this analysis, all given statements about the parabola are false in a general sense. There are specific cases where some could be true (for example, a parabola that only touches the x-axis would make the second statement true), but without specific conditions, they do not hold.
If you have a specific parabola in mind or specific conditions, please provide them for a more accurate assessment!
1 answer