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Determine which of the following statements is true about the parabola.

(1 point)
Responses

The y-intercept and the vertex share the same point.
The y -intercept and the vertex share the same point.

There is only one x-intercept.
There is only one x -intercept.

The y-intercept and the x-intercepts share the same point.
The y -intercept and the x -intercepts share the same point.

The vertex and the x-intercepts share the same point.

1 answer

To determine which statement is true about a parabola, we need to clarify what each term refers to.

  1. Y-intercept: This is the point where the parabola crosses the y-axis (where \(x = 0\)). It is found by evaluating the function at \(x = 0\).

  2. Vertex: This is the highest or lowest point on the parabola, which is determined by the formula for the vertex based on the coefficients of the quadratic equation (if in standard form \(y = ax^2 + bx + c\), the x-coordinate of the vertex is given by \(x = -\frac{b}{2a}\)).

  3. X-intercepts: These are the points where the parabola crosses the x-axis (where \(y = 0\)). The number of x-intercepts can vary based on the parabola's orientation and position.

Now let's evaluate the statements:

  • "The y-intercept and the vertex share the same point." This is only true in specific cases where the vertex lies on the y-axis (for example, in the case of a parabola that is symmetric around the y-axis and faces upwards or downwards).

  • "There is only one x-intercept." This is true for parabolas that touch the x-axis at one point (the vertex is on the x-axis), but not all parabolas have this property.

  • "The y-intercept and the x-intercepts share the same point." This is false. The y-intercept occurs when \(x = 0\), while x-intercepts occur when \(y = 0\). They are typically different points unless the parabola specifically passes through the point (0,0), which is also the origin.

  • "The vertex and the x-intercepts share the same point." This can also be true in specific instances where the vertex is on the x-axis. However, in general, they do not share the same point.

Based on this analysis, none of the statements are universally true for all parabolas. The statements could occasionally be true in specific cases or types of parabolas, but there isn't a definitive true statement applicable to all parabolas as provided.

If we must pick a statement, it would be contextual based on the specific properties of certain parabolas. Can you provide more details about the parabola in question for better clarity?