What is the equation for the graph shown below?

Try this way:
the common view of the required equation is y=(x-a)(x-b), where 'a' and 'b' are the interception points (the curve and Ox).
using the coordinates of interception points it is possible to make up the required equation: x=3 and x=5.
y=(x-3)(x-5).
If to check the equation using the coordinates of the vertex (4;-1):
-1=(4-3)(4-5) ⇔ -1=-1, it means, that the equation made up is correct.

y=(x-3)(x-5) or y=x²-8x+15.