Final answer:
To determine the coefficients a, b, and c for the quadratic equation y = ax² + bx + c that goes through the points (-6,15), (0,-3), and (2,7), we set up and solve a system of equations using these points. This leads to the unique solution for a, b, and c.
Step-by-step explanation:
To find the values of a, b, and c in the quadratic equation y = ax² + bx + c, we can plug in the given coordinates and solve the system of equations that we get. The coordinates given are (-6, 15), (0, -3), and (2, 7).
First, using the coordinate (0, -3):
-3 = a(0)^2 + b(0) + c
So, c = -3.
Using the coordinate (-6, 15):
15 = a(-6)^2 + b(-6) + c
15 = 36a - 6b - 3
Using the coordinate (2, 7):
7 = a(2)^2 + b(2) + c
7 = 4a + 2b - 3
Now you have a system of equations:
1) 36a - 6b = 18
2) 4a + 2b = 10
By solving this system, you can find the values of a and b that satisfy both equations.