SOLUTION: Given a>0 for what values of k does y=a(x-k)^2+k have? a) two x-intercepts ? b) one x intercept ? c) no x-intercepts? Please explain why for each one. Thank you.

Question 1003127: Given a>0 for what values of k does y=a(x-k)^2+k have?
a) two x-intercepts ?
b) one x intercept ?
c) no x-intercepts?
Please explain why for each one. Thank you.
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
The x-intercepts, if any, are the points where (or the values of for those points).

IF YOU WERE NOT TAUGHT ABOUT THE DISCRIMINANT OF A QUADRATIC EQUATION:
The x-intercepts, if any, are the points where (or the values of for those points).
For the function is , and has one x-intercept at
For , since for all values of ,
, meaning that there is no x-intercept.
For , there are two intercepts, because , so , and
-->-->-->-->--> .

IF YOU WERE TAUGHT ABOUT THE DISCRIMINANT OF A QUADRATIC EQUATION:
The reasoning above is still valid, but you may be expected to talk about the discriminant in an answer.
That makes it complicated, but here it goes:
<---><---><--->
For a quadratic equation of the form ,
.
If , has two solutions;
if , has one solution, and
if , has no solution.
Applied to ,
where and ,
, so
if <--> , has two solutions, and has two x-intercepts;
if <--> , has one solutions, and has one x-intercepts, and
if <--> , has no solutions, and has no x-intercepts.

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