Final answer:
To find the equation of the parabola with vertex at (5,-3) and passing through (2,-1), we use the vertex form y = a(x - h)^2 + k and solve for 'a'. The equation is y = (2/9)(x - 5)^2 - 3. The y-coordinate at the y-axis intersection is 23/9.
Step-by-step explanation:
To determine an equation for the parabola with a vertex at the point (5,-3) and containing the point (2,-1), we use the vertex form of a parabola which is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. Plugging in the vertex, we get y = a(x - 5)^2 - 3. We can then insert the point (2, -1) into the equation to solve for the coefficient 'a'.
Substituting the coordinates of the point (2, -1) into the equation gives us -1 = a(2 - 5)^2 - 3. Simplifying, we get -1 = 9a - 3. Adding 3 to both sides gives 2 = 9a, and solving for 'a' gives a = 2/9. Therefore, the equation of the parabola is y = (2/9)(x - 5)^2 - 3.
To find the y-coordinate where the parabola intersects the y-axis, we set x to 0 and solve for y. In this case, y = (2/9)(0 - 5)^2 - 3 which simplifies to y = (2/9)(25) - 3. Thus, y = 50/9 - 3. Converting 3 to a fraction with 9 as the denominator, we get 27/9, and subtracting from 50/9 yields y = 23/9. Therefore, the y-coordinate where the parabola intersects the y-axis is 23/9.