Questions: The vertex of a parabola is in the first quadrant of a coordinate grid. A line with a negative slope passes through the origin. If the parabola and line intersect at the origin, which statement must be true? The parabola opens downward. The parabola opens upward. The slope of the line is equal to -1 . The slope of the line is not equal to -1 .

The vertex of a parabola is in the first quadrant of a coordinate grid. A line with a negative slope passes through the origin. If the parabola and line intersect at the origin, which statement must be true?

The parabola opens downward.
The parabola opens upward.
The slope of the line is equal to -1 .
The slope of the line is not equal to -1 .

Transcript text: The vertex of a parabola is in the first quadrant of a coordinate grid. A line with a negative slope passes through the origin. If the parabola and line intersect at the origin, which statement must be true? The parabola opens downward. The parabola opens upward. The slope of the line is equal to -1 . The slope of the line is not equal to -1 .

Solution

Solution Steps

To determine which statement must be true, we need to consider the geometric properties of the parabola and the line. Since the vertex of the parabola is in the first quadrant and the parabola intersects the origin, it must open downward to intersect the origin from the first quadrant. The line with a negative slope passing through the origin can have any negative slope, so the slope is not necessarily -1.

Step 1: Analyze the Parabola's Orientation

Given that the vertex of the parabola is in the first quadrant and it intersects the origin, the parabola must open downward. This is because a parabola with a vertex in the first quadrant that opens upward would not intersect the origin.

Step 2: Analyze the Line's Slope

The line passes through the origin and has a negative slope. This means the line can have any negative slope, not necessarily \(-1\). Therefore, the statement that the slope of the line is equal to \(-1\) is false, and the statement that the slope is not equal to \(-1\) is true.