How many intersections are there between the parabola y = x ^ 2-ax + A-2 and the coordinate axis I really think it's three, but the teacher said no, she said three wrong, help me!

y=x^2-ax+a-2=(x-a/2)²-a²/4+a-2
The parabola has the minimum value - A & # / 4 + A-2, with the opening upward
∵-a²/4+a-2
=-1/4*（a²-4a+8）
=-1/4*[(a²-4a+4)+4]
=-1 / 4 * (A-2) & 178; - 1 ≤ - 1
There are two intersections with the x-axis
And ∵ has an intersection with the y-axis
There are three intersections with the coordinate axis
Special case: when the parabola passes through the origin, there are only two intersections
That is, when x = 0, y = 0, then a = 2

The number of intersections of parabola y = x ^ 2-ax + A-2 and coordinate axis is ()

(1)
The number of intersections of y = x ^ 2-ax + A-2 and X axis is the number of solutions of X when y = 0
x^2-ax+a-2=0
Discriminant = a ^ 2-4 * (A-2) = a ^ 2-4a + 8 = (A-2) ^ 2 + 4 ≥ 4 > 0
There are two intersections with the x-axis
(2)
The number of intersections of y = x ^ 2-ax + A-2 and Y axis is the number of solutions of y when x = 0
y=0-0+a-2=a-2
There is an intersection with the y-axis
There are three intersections with the two axes
Number of intersections with coordinate axis

How many intersections are there between the parabola y = - x square + 2x-3 and the coordinate axis

y=-x²+2x-3
=-(x²-2x+3)
=-[(x-1)²+2]
=-(x-1)²-2
The maximum value of the function is - 2
Function image and X-axis unfocused

It is known that the parabola y is equal to x plus MX minus 2m, M is not equal to zero one

What's the problem

How many intersections are there between the parabola y = x ^ 2-ax + A-2 and the coordinate axis I really think it's three, but the teacher said no, she said three wrong, help me!