Curve Fitting Find the function $y=a{x}^2+bx+c$whose graph contains the points $\left(1,-1\right),\left(3,-1\right)$and $\left(-2,14\right)$.

The given points are$\left(1,-1\right),\left(3,-1\right)$and$\left(-2,14\right)$.

We require that the three points satisfy the equation$y=a{x}^2+bx+c$.

For the point $\left(1,-1\right)$:

$$\begin{gathered} -1=a{\left(1\right)}^2+b\left(1\right)+c \\ -1=a+b+c \end{gathered}$$

For the pointrole="math" localid="1648222565949" $\left(3,-1\right)$:

$$\begin{gathered} -1=a{\left(3\right)}^2+b\left(3\right)+c \\ -1=9a+3b+c \end{gathered}$$

For the point $\left(-2,14\right)$:

$$\begin{gathered} 14=a{\left(-2\right)}^2+b\left(-2\right)+c \\ 14=4a-2b+c \end{gathered}$$

Now, determine $a$, $b$and $c$so that each equation is satisfied.

Solve the following system of three equations containing three variables:

$$\begin{gathered} a+b+c=-1...\left(1\right) \\ 9a+3b+c=-1...\left(2\right) \\ 4a-2b+c=14...\left(3\right) \end{gathered}$$

From equation (1) and (2),

$(-)a+b+c=-19a+3b+c=-1-8a-2b=0$

$$\begin{gathered} -8a=2b \\ b=-4a...\left(4\right) \end{gathered}$$

From equation (2) and equation (3),

$(-)9a+3b+c=-14a-2b+c=145a+5b=-15$

role="math" localid="1648223323515" $a+b=-3...\left(5\right)$

Substituting value of $b$in the equation (5),

role="math" localid="1648223877636" $$\begin{gathered} a-4a=-3 \\ -3a=-3 \\ a=1 \end{gathered}$$

From equation (4),

role="math" localid="1648223891857" $$\begin{gathered} b=-4\left(1\right) \\ b=-4 \end{gathered}$$

Substituting values of $a$and $b$in the equation (1),

role="math" localid="1648223927139" $$\begin{gathered} 1-4+c=-1 \\ c=2 \end{gathered}$$

So, the quadratic function whose graph contains the points $\left(1,-1\right),\left(3,-1\right)$and $\left(-2,14\right)$is

role="math" localid="1648223944701" $y=x^2-4x+2$

This is the equation of the curve.

The graph will be