Determine which of the following statements is true about the parabola.
A. The y-intercept and the vertex share the same point.
B. The y-intercept and the x-intercepts share the same point.
C. The vertex and the x-intercepts share the same point.
D. There is only one x-intercept.
C. The vertex and the x-intercepts share the same point.
To determine which statement is true about the parabola, let's review the properties of a parabola.
A parabola is a U-shaped curve that can open upwards or downwards. It is defined by its vertex, axis of symmetry, and the points where it intersects the x-axis (x-intercepts) and y-axis (y-intercept).
A. The statement "The y-intercept and the vertex share the same point" is not true. The y-intercept is the point where the parabola intersects the y-axis, while the vertex is the highest or lowest point on the parabola.
B. The statement "The y-intercept and the x-intercepts share the same point" is not true either. The y-intercept is a single point on the y-axis, while the x-intercepts are the points where the parabola intersects the x-axis.
C. The statement "The vertex and the x-intercepts share the same point" is not true either. The vertex is a single point on the axis of symmetry, whereas the x-intercepts are points on the x-axis.
D. The statement "There is only one x-intercept" is true for some parabolas. A parabola can have one, two, or no x-intercepts depending on its orientation and position relative to the x-axis.
Therefore, the correct statement is D, "There is only one x-intercept."
To determine which statement is true about the parabola, we need to understand the properties of a parabola.
A parabola is a U-shaped curve that can either open upwards or downwards. It has a vertex, which is the highest or lowest point on the curve, and it may intersect the x-axis at one or two points, known as the x-intercepts. The y-intercept is the point where the parabola intersects the y-axis.
Now, let's analyze each statement:
A. The y-intercept and the vertex share the same point.
This statement is not generally true for all parabolas. The y-intercept is the point where the parabola intersects the y-axis, while the vertex is the highest or lowest point on the curve. In most cases, the vertex and the y-intercept are not the same point. Therefore, this statement is most likely false.
B. The y-intercept and the x-intercepts share the same point.
The y-intercept is the point where the parabola intersects the y-axis, while the x-intercepts are the points where it intersects the x-axis. These two intercepts occur at different locations along the graph of the parabola. Therefore, this statement is false.
C. The vertex and the x-intercepts share the same point.
The vertex is the highest or lowest point on the parabola, while the x-intercepts are where it intersects the x-axis. Generally, the vertex does not coincide with the x-intercepts. However, there is a special case where the vertex and the x-intercepts can coincide, resulting in a parabola with only one x-intercept. Therefore, this statement could be true.
D. There is only one x-intercept.
This statement is true for a specific case mentioned above, where the parabola has a single x-intercept. In this scenario, the parabola opens upwards or downwards but does not intersect the x-axis at any other point except one.
Based on the explanations, the statement that is most likely true about the parabola is D. There is only one x-intercept.