Question

Determine which of the following statements is true about the parabola.
A) The vertex and the x-intercepts share the same point.
B) There is only one x-intercept.
C) The y-intercept and the x-intercepts share the same point.
D) The y-intercept and the vertex share the same point.

Answer 1

To determine which statement is true about a parabola, let's analyze each option.

A) **The vertex and the x-intercepts share the same point.**
- This is true only for a special case where the parabola touches the x-axis at the vertex (i.e., the vertex is the turning point and the only x-intercept). This is not true for all parabolas.

B) **There is only one x-intercept.**
- This can be true for parabolas that are tangent to the x-axis (i.e., they have a double root). However, many parabolas have two x-intercepts or none at all. So this is not universally true.

C) **The y-intercept and the x-intercepts share the same point.**
- The y-intercept occurs when \( x = 0 \), while x-intercepts occur when \( y = 0 \). The y-intercept and x-intercepts will generally be different points unless the parabola passes through the origin which is not true for all parabolas.

D) **The y-intercept and the vertex share the same point.**
- This is only true for specific cases, such as when the vertex is at the point of the y-intercept. However, this is not true for all parabolas.

Conclusion: In general, none of the statements are universally true for all parabolas. Instead, they might only hold for particular parabolas.

If we specifically consider a general parabola, none of the statements can be categorized as true in all cases. Therefore, among the options given, there is no statement that universally holds for all parabolas.