Question

Find the equation of the parabola whose axis is parallel to x-axis and passing through $(-2,1),(1,2),(-1,3)$

Answer 1

Equation of parabola whose axis is parallel to x-axis in general is $y=a{x}^2+bx+c$

Substituting $(-2,1)$

$1=a\left(4\right)+b(-2)+c$

$4a-2b+c=1$...(I)

Substituting $(1,2)$

$2=a(1{)}^2+b(1)+c$

$a+b+c=2$.....(ii)

Substituting $(-1,3)$

$3=a\left(1\right)+b(-1)+c$

$a-b+c=3$....(iii)

$c=3-a+b$...(iv)

Substituting (iv) in (i) and (iii)

$4a-2b+3-a+b=1$

$3a-b=-2$....(v)

$a+b+3-a+b=2$

$2b=-1$

$b=\frac{-1}{2}$

Replacing value of $b$ in (v)

$3a+\frac{1}{2}=-2$

$3a=\frac{-5}{2}$

$a=-\frac{-5}{6}$

$c=3-\left(\frac{-5}{6}\right)+\left(\frac{-1}{2}\right)$

$c=\frac{10}{3}$

$y=\frac{-5}{6}{x}^2-\frac{1}{2}+\frac{10}{3}$.