Question

use quadratic regression to find a function that fits the following points. (-1, -15) (1, -7) (6, -22) ?x² + x +

Explanation:

Step1: Set up the quadratic function

Let the quadratic function be $y = ax^{2}+bx + c$. Substitute the given points into the function.
For the point $(-1,-15)$: $-15=a(-1)^{2}+b(-1)+c=a - b + c$.
For the point $(1,-7)$: $-7=a(1)^{2}+b(1)+c=a + b + c$.
For the point $(6,-22)$: $-22=a(6)^{2}+b(6)+c=36a+6b + c$.

Step2: Create a system of equations

We have the system of equations:

$$\begin{cases}a - b + c=-15\\a + b + c=-7\\36a+6b + c=-22\end{cases}$$

Subtract the first - equation from the second equation:
$(a + b + c)-(a - b + c)=-7-(-15)$
$2b = 8$, so $b = 4$.

Step3: Substitute $b = 4$ into the equations

Substitute $b = 4$ into the first and third equations:
First equation becomes $a-4 + c=-15$, or $a + c=-11$.
Third equation becomes $36a+6\times4 + c=-22$, or $36a + c=-46$.

Step4: Solve for $a$

Subtract the new - first equation from the new - third equation:
$(36a + c)-(a + c)=-46-(-11)$
$35a=-35$, so $a=-1$.

Step5: Solve for $c$

Substitute $a=-1$ into $a + c=-11$, we get $-1 + c=-11$, so $c=-10$.

Answer:

$-1x^{2}+4x - 10$