Question

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.

Explanation:

1. Consider the general form of a parabola \(y = a(x - h)^2 + k\) with vertex \((h,k)\) in the first - quadrant (\(h>0,k > 0\)) and a line \(y=mx\) (\(m<0\)) passing through the origin \((0,0)\).
2. If the parabola opens upward (\(a>0\)), as \(x\) moves away from \(0\) in the negative \(x\) - direction, the parabola will go down from the vertex but will still be above the \(x\) - axis near the origin. As \(x\) moves away from \(0\) in the positive \(x\) - direction, the parabola will go up. A line with a negative slope \(y = mx\) (\(m<0\)) passing through the origin will intersect an upward - opening parabola at the origin.
3. If the parabola opens downward (\(a < 0\)), starting from the vertex in the first - quadrant, the parabola will go down in both positive and negative \(x\) - directions. It may not intersect a line with a negative slope passing through the origin at the origin only (it could have other intersection points or not intersect at all depending on the slope and the position of the vertex).
4. The slope of the line \(m\) can be any negative number. Just because the line has a negative slope and passes through the origin does not mean \(m=-1\).

Answer:

The parabola opens upward.