The equation of which graph has a negative discriminant?

Question

The equation of which graph has a negative discriminant?

A. The graph shows a parabola opening up with a vertex at 2 comma negative 1
 
B. The graph shows a parabola opening up with a vertex at negative 1 comma negative 3
 
C. The graph shows a parabola opening down with a vertex at negative 1 comma negative 1
 
D. The graph shows a parabola opening down with a vertex at negative 1 comma 0

Answer 1

If the graph intersects or touches the x-axis, it means that there is at least one real solution when setting the function equal to 0.  The solution occurs at that point.

 

If there are any real solutions, the discriminant must be zero or positive.  But if the graph does NOT touch the x-axis, then there are no real solutions, which means that the discriminant is negative.

 

So you are looking for a graph that does NOT touch the x-axis.

 

A has a y value less than zero (-1) and opens up, so it must cross the x-axis.

 

The same is true for B, which has a y value at the vertex of -3.

 

D intersects the x-axis at its vertex, (-1, 0)

 

However, C's vertex has a y value of -1 and it opens down, so all other y values must be less than -1.  So C does not intersect the x-axis at all.  So the answer is C