SOLUTION: Without drawing a graph how can I determine (a) how many x-intercepts the parabola has (b) whether its vertex is above, below or on the x- axis? y = -x^2 + 2x-1

Algebra ->  Graphs -> SOLUTION: Without drawing a graph how can I determine (a) how many x-intercepts the parabola has (b) whether its vertex is above, below or on the x- axis? y = -x^2 + 2x-1      Log On


   

Question 7460: Without drawing a graph how can I determine (a) how many x-intercepts the parabola has (b) whether its vertex is above, below or on the x- axis?

y = -x^2 + 2x-1
Answer by ichudov(507) About Me  (Show Source):
You can put this solution on YOUR website!
x intercepts is just another way of saying "roots". Find roots of this quadratic expression with y set to 0.
since coefficient in front of x is negative, the parabola is pointing downwards.
where its vertex is, depends on the roots.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case -1x%5E2%2B2x%2B-1+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%282%29%5E2-4%2A-1%2A-1=0.

Discriminant d=0 is zero! That means that there is only one solution: x+=+%28-%282%29%29%2F2%5C-1.
Expression can be factored: -1x%5E2%2B2x%2B-1+=+-1%28x-1%29%2A%28x-1%29

Again, the answer is: 1, 1. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-1%2Ax%5E2%2B2%2Ax%2B-1+%29


since both roots are same, its vertex is right on the x axis.