what is a quadratic equation to model this table:

Question

x y

1 1

2 7

3 17

Answer 1

The quadratic equation is y = ax² + bx + c written in standard form where a, b, and c are the leading coefficients in each term.

We need at least 3 rows in the table since there are 3 variables (a, b, and c).

We have the following points ⇒ (x, y): (1, 1), (2, 7), and (3, 17).

We have a set of three simultaneous equations:

(1) 1 = a(1)² + b(1) + c = a + b + c

(2) 7 = a(2)² + b(2) + c = 4a + 2b + c

(3) 17 = a(3)² + b(3) + c = 9a + 3b + c

Subtract (1) from (2) to get the new equation. Eliminate the c's.

(4) 7 - 1 = (4a + 2b + c) - (a + b + c) ⇒ 6 = 3a + b

Subtract (2) from (3) to get the new equation. Eliminate the c's.

(5) 17 - 7 = (9a + 3b + c) - (4a + 2b + c) ⇒ 10 = 5a + b

Subtract (4) from (5) to get the new equation. Eliminate the b's.

(6) 10 - 6 = (5a + b) - (3a + b) ⇒ 4 = 2a ⇒ a = 2

Substitute a = 2 in (4) to find b ⇒ 3(2) + b = 6 ⇒ 6 + b = 6 ⇒ b = 0

Substitute a = 2 and b = 0 in (1) to find c ⇒ 2 + 0 + c = 1 ⇒ 2 + c = 1 ⇒ c = -1

Quadratic equation: y = 2x² + 0x - 1 ⇒ y = 2x² - 1