Question

complete the statements.
graph □ has one real root.
graph □ has a negative discriminant.
graph □ has an equation with coefficients
a = 1, b = 4, c = -2
done ✓

Explanation:

Step1: Analyze real roots (x-intercepts)

A parabola with one real root touches the x - axis at one point (vertex on x - axis). Looking at the graphs, Graph B touches the x - axis at one point, so Graph B has one real root.

Step2: Analyze negative discriminant

A negative discriminant means no real roots (parabola doesn't intersect x - axis). Graph A is above the x - axis (no x - intercepts), so Graph A has a negative discriminant.

Step3: Analyze equation \(y = ax^{2}+bx + c\) with \(a = 1,b = 4,c=-2\)

First, find the vertex of \(y=x^{2}+4x - 2\). The x - coordinate of the vertex is \(-\frac{b}{2a}=-\frac{4}{2\times1}=-2\). The y - coordinate is \(y=(-2)^{2}+4\times(-2)-2=4 - 8 - 2=-6\). The parabola opens upwards (\(a = 1>0\)) and has vertex at \((-2,-6)\). Graph C matches this description (vertex lower, opens up, and we can check the shape and position).

Answer:

1. Graph B has one real root.
2. Graph A has a negative discriminant.
3. Graph C has an equation with coefficients \(a = 1,b = 4,c=-2\)