Final answer:
To determine b and c, use the vertex form of a quadratic equation, plug in the vertex coordinates and the coefficient a, expand, and compare with the original equation.
Step-by-step explanation:
To find the values of b and c for the quadratic equation y = 13x² + bx + c given that the vertex is at (9, 4), we use the vertex form of a quadratic equation: y = a(x - h)² + k, where (h, k) is the vertex of the parabola and a is the coefficient of x².
Since we know that a=13 and the vertex (h, k)=(9, 4), we substitute these into the vertex form and get: y = 13(x - 9)² + 4. Expanding this, we get: y = 13(x² - 18x + 81) + 4 = 13x² - 234x + 1053 + 4 = 13x² - 234x + 1057.
Comparing this with the original form, we can say that b = -234 and c = 1057.