Question

14) Consider the parabola with equation y = x^2 - 6x + 5.
a. Use any suitable method to determine the coordinates of the turning point of this parabola.
b. Hence, state for which values of c the line y = c will intersect the parabola:
i. twice
ii. once
iii. not at all

Answer 1

You are probably expected to use "completing the square", although other methods are just as suitable.

y = x^2 - 6x + 5
= x^2 - 6x + 9 - 9 + 5
= (x-3)^2 - 4

The vertex is (3, -4)
so a horizontal line of y = -4 will run through this vertex , that's your "once"
y = -3 will cut it twice
y = -6 will not touch the parabola

in general
y = -4 , once
y = c , c > -4, twice
y = c c < -4 , not at all