Final answer:
To find the equation of a parabola that passes through a given point and has a vertex, we can use the standard form of the equation for a parabola. By substituting the vertex and the given point into the equation, we can solve for the value of 'a'. The equation can then be written in the form y = ax^2 + bx + c.
Step-by-step explanation:
To find the equation of a parabola that passes through the point (4,-7) and has a vertex at (1,-6), we can use the standard form of the equation for a parabola, which is y = a(x-h)^2 + k. In this equation, (h,k) represents the coordinates of the vertex. Substituting the values of the vertex, we get: y = a(x-1)^2 - 6. Now, we can use the given point (4,-7) to find the value of 'a'. Substituting these values into the equation, we get: -7 = a(4-1)^2 - 6. Simplifying further, we have: -7 = 9a - 6. Solving this equation, we find that 'a' is equal to -1.
Therefore, the equation of the parabola that passes through (4,-7) and has a vertex at (1,-6) is y = -1(x-1)^2 - 6.