To find a quadratic function of the form [tex]\( f(x) = ax^2 + bx + c \)[/tex] that fits the points [tex]\((-3, 117)\)[/tex], [tex]\((0, 45)\)[/tex], and [tex]\((3, 27)\)[/tex], we can use the method of quadratic regression. Here are the steps to find the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
1. Set up the system of equations:
Each point [tex]\((x, y)\)[/tex] gives us an equation of the form [tex]\( y = ax^2 + bx + c \)[/tex].
For the point [tex]\((-3, 117)\)[/tex]:
[tex]\[
117 = a(-3)^2 + b(-3) + c \implies 117 = 9a - 3b + c
\][/tex]
For the point [tex]\((0, 45)\)[/tex]:
[tex]\[
45 = a(0)^2 + b(0) + c \implies 45 = c
\][/tex]
For the point [tex]\((3, 27)\)[/tex]:
[tex]\[
27 = a(3)^2 + b(3) + c \implies 27 = 9a + 3b + c
\][/tex]
2. Substitute [tex]\(c = 45\)[/tex] from the second equation into the other two equations:
Substituting [tex]\(c = 45\)[/tex] into the first equation:
[tex]\[
117 = 9a - 3b + 45 \implies 72 = 9a - 3b \implies 24 = 3a - b
\][/tex]
Substituting [tex]\(c = 45\)[/tex] into the third equation:
[tex]\[
27 = 9a + 3b + 45 \implies -18 = 9a + 3b \implies -6 = 3a + b
\][/tex]
3. Solve for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
We now have a system of linear equations:
[tex]\[
24 = 3a - b \quad \text{(Equation 1)}
\][/tex]
[tex]\[
-6 = 3a + b \quad \text{(Equation 2)}
\][/tex]
Add Equation 1 and Equation 2:
[tex]\[
24 + (-6) = 3a - b + 3a + b \implies 18 = 6a \implies a = 3
\][/tex]
Substitute [tex]\(a = 3\)[/tex] back into Equation 1:
[tex]\[
24 = 3(3) - b \implies 24 = 9 - b \implies b = -15
\][/tex]
4. Determine the coefficients:
We have found the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] as follows:
[tex]\[
a = 3, \quad b = -15, \quad c = 45
\][/tex]
5. Write the quadratic function:
Therefore, the quadratic function that fits the points [tex]\((-3, 117)\)[/tex], [tex]\((0, 45)\)[/tex], and [tex]\((3, 27)\)[/tex] is:
[tex]\[
f(x) = 3x^2 - 15x + 45
\][/tex]
So, the quadratic function that models the given points is:
[tex]\[ f(x) = 3x^2 - 15x + 45 \][/tex]
