Question

Write the equation of the quadratic function that passes through the points (-1, 1), (1, 5), and (2,10).

Answer 1

`\displaystyle f(x)=x^2+2x+2`

Explanation:

System Of Linear Equations

In this problem, we'll need to solve a 3x3 system of linear equations because we have three unknowns and three conditions.

We are required to find the equation of the quadratic function that passes through the points (-1, 1), (1, 5), and (2,10)

The general quadratic function can be written as

`\displaystyle f(x)=ax^2+bx+c`

We need to find the values of a,b, and c. Let's use the first condition, i.e. f(-1)=1

`\displaystyle f(-1)=a(-1)^2+b(-1)+c`

`\displaystyle f(-1)=a-b+c`

`\displaystyle a-b+c=1.....[eq\ 1]`

Now we use the second condition f(1)=5

`\displaystyle f(1)=a(1)^2+b(1)+c`

`\displaystyle f(1)=a+b+c`

`\displaystyle a+b+c=5.......[eq\ 2]`

Finally, we use the third condition f(2)=10

`\displaystyle f(2)=a(2)^2+b(2)+c`

`\displaystyle f(2)=4a+2b+c`

`\displaystyle 4a+2b+c=10....[eq\ 3]`

We put together eq 1, eq 2, and eq 3 to form the system

`\displaystyle \left\{\begin{matrix}a-b+c=1\\ a+b+c=5\\ 4a+2b+c=10\end{matrix}\right.`

Adding the first two equations we have

`\displaystyle 2a+2c=6`

`\displaystyle a+c=3`

And also

`\displaystyle b=2`

Using the above equation and the value of b in the third equation, we have

`\displaystyle \left\{\begin{matrix}a+c=3\\ 4a+c=6\end{matrix}\right.`

Subtracting the first equation from the second

`\displaystyle 3a=3`

`\displaystyle a=1`

And therefore

`\displaystyle c=2`

Now we have all the values, the quadratic function is

`\displaystyle \boxed{f(x)=x^2+2x+2}`