Without drawing the graph of the equation, determine how many points the parabola has in common with the x-axis and whether its vertex lies above, on, or below the x-axis.

y = –2x2 + x + 3

Question

Without drawing the graph of the equation, determine how many points the parabola has in common with the x-axis and whether its vertex lies above, on, or below the x-axis.

y = –2x2 + x + 3

anonymous · Answer

Answers could be
1 point in common; vertex on x-axis
       2 points in common; vertex below x-axis
       2 points in common; vertex above x-axis
       no points in common; vertex below x-axis

debbieg · Answer

the # points in common can be determined from the discriminant.

D=0, 1 real root, so 1 point in common.
D>0, 2 real roots, so 2 points in common
D<0, no real roots so no points in common

If 2 intercepts or no intercepts, the direction of opening will tell you where the vertex lies.

2 intercepts and opens up means that the vertex lies below axis.
no intercepts and opens up means that the vertex lies above axis.
....etc.

Just picture it in your head, can you see why these must be true?

anonymous · Answer

Okay, It opens up going down so there for it would be 2 points in common, vertex below the x aixs. correct?

anonymous · Answer

@DebbieG

debbieg · Answer

Well, without even going back up to look at the equation, I can tell you that something is wrong. Think about it: if the vertex is BELOW the x-axis and the parabola opens DOWN, there certainly can't be 2 points of intersection with the axis, right?

debbieg · Answer

So your equation is $$y = –2x^2 + x + 3 $$
Since a=-2<0, you are right in that it opens down.

The discriminant is $$D=1^2-4(-2)(3)=1+24=25$$

debbieg · Answer

So what does that tell you about the number of real solutions?