At what values of a and c, the graph of the function y = ax ^ 2-2x + c passes through the points A (1; 6) and B (2; 19).

A point lies on a curve if, when substituting its argument into the equation of the curve, we obtain the value of its ordinate. In order for both points to lie on a given curve, you need to solve a system of equations with their arguments.

Point A (1; 6):

6 = 1 * a – 2 * 1 + c, whence:

a + c = 8 and c = 8-a;

Point B (2; 19):

19 = (2) ^ 2 – 2 * 2 + c, whence
4a + c – 4 = 19;

4a + c = 23;

Substitute the value of c obtained from their equation for point A:

4a + 8 -a = 23;

3a = 15;

a = 5, then c = 3.

The equation takes the form y = 5x ^ 2 – 2x +3.

Answer: a = 5; c = 3