Answer:
-79.75
Step-by-step explanation:
You want the value of c that makes y=c intersect the parabola y = -x² +6x -100 in exactly one place.
Horizontal line
A horizontal line will intersect a parabola in one place if it is tangent to the parabola at its vertex. The vertex of the given parabola can be found by writing its equation in vertex form:
y = -(x² -9x) -100
We can add and subtract 4.5² to complete the square:
y = -(x -4.5)² -100 +20.25
y = -(x -4.5)² -79.75
Comparing this to vertex form ...
y = a(x -h)² +k
we see that (h, k) = (4.5, -79.75). These are the coordinates of the vertex.
A horizontal line will be tangent to the vertex it its equation is y = -79.75.
The value of c is -79.75.
