Final answer:
The given point (1,2) lies on the parabola with center at (1,-4) and vertex at (7,-4).
Step-by-step explanation:
To determine if a given point lies on a parabola, we can use the equation of the parabola.
Given that the center of the parabola is (1, -4) and the vertex is (7, -4), we can use the equation of a parabola in vertex form: y = a(x - h)^2 + k, where (h, k) is the vertex. Plugging in the values, we have: y = a(x - 7)^2 - 4.
Now we can substitute the coordinates of the given point, (1, 2), into the equation to check if it satisfies the equation.
2 = a(1 - 7)^2 - 4
2 = a(-6)^2 - 4
2 = a(36) - 4
2 = 36a - 4
36a = 6
a = 6/36 = 1/6
So, the equation of the parabola is y = (1/6)(x - 7)^2 - 4. Therefore, the given point (1, 2) does lie on the parabola.