Final answer:
The correct equation of the parabola with the given x-intercept, y-intercept, and vertex is Option A: y = (x - 4)^2 - 3. The value of 'a' is found by substituting the x-intercept into the vertex form of the equation, yielding a positive coefficient and an upward-facing parabola.
Step-by-step explanation:
The equation of a parabola with a vertex at (4, -3) and the symmetry axis being vertical can be written in the form y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. Since the vertex is given as (4, -3), we substitute h and k to get the equation y = a(x - 4)^2 - 3. Now, given that one x-intercept is at (8, 0), we can plug these values into our equation to find the value of a. We get:
0 = a(8 - 4)^2 - 3
0 = 16a - 3
16a = 3
a = 3/16
Because the coefficient a is positive, the parabola opens upwards, and since it includes the y-intercept (0,0), the equation could not have a negative coefficient before the squared term, as it would not cross the y-axis there.
Thus, the correct equation of the parabola is Option A: y = (x - 4)^2 - 3, which can be written as y = (3/16)(x - 4)^2 - 3 when the value of 'a' is included.