Question

write an equation of the parabola that passes through the points (0, -9), (-2, 1), and (1, 4). write your answer in the form y = ax² + bx + c, where a, b, and c are integers

Explanation:

Step1: Substitute points into equation

Substitute $(0, - 9)$ into $y = ax^{2}+bx + c$:
$c=-9$
Substitute $(-2,1)$ into $y = ax^{2}+bx + c$:
$4a-2b + c=1$
Substitute $(1,4)$ into $y = ax^{2}+bx + c$:
$a + b + c=4$

Step2: Substitute $c = - 9$ into other two - equations

Substitute $c=-9$ into $4a-2b + c=1$:
$4a-2b-9 = 1$, simplifies to $4a-2b=10$, then $2a - b=5$ (divide by 2)
Substitute $c=-9$ into $a + b + c=4$:
$a + b-9 = 4$, simplifies to $a + b=13$

Step3: Solve the system of equations

Add the equations $2a - b=5$ and $a + b=13$:
$(2a - b)+(a + b)=5 + 13$
$3a=18$
$a = 6$
Substitute $a = 6$ into $a + b=13$:
$6 + b=13$
$b=7$

Answer:

$y = 6x^{2}+7x-9$