\(\text{Find all numbers $a$ for which the graph of $y=x^2+a$ and the graph of $y=ax$ intersect. Express your answer in interval notation. }\)
y = x^2 + a
y = ax
Setting the y's equal
x^2 + a = ax rearrange as
x^2 - ax + a = 0
For us to have real solutions.....the discriminant must be ≥ 0
So
a^2 - 4a ≥ 0
a ( a - 4) ≥ 0
This will be true on these intervals
a ≥ 4 and a ≤ 0
(-infinity , 0 ] U [4, infinity )