Question

1
select the correct answer from each drop-down menu.
the annual enrollment of a university for the last nine years is recorded in the table.
year 1 2 3 4 5 6 7 8 9
students (thousands) 9.5 8 8.5 7.5 6.5 6.5 8.5 8.5 9
what quadratic function best models this set of data?
$y = \boldsymbol{\square} x^2 + \boldsymbol{\square} x + \boldsymbol{\square}$

Explanation:

Step1: Define quadratic form

A quadratic function has the form $y = ax^2 + bx + c$, where $a$, $b$, $c$ are constants to solve for. We use 3 data points from the table: $(1, 9.5)$, $(5, 6.5)$, $(9, 9)$ to set up a system of equations.

Step2: Substitute $(1, 9.5)$

Substitute $x=1$, $y=9.5$ into the quadratic formula:
$a(1)^2 + b(1) + c = 9.5$
$\implies a + b + c = 9.5$ --- Equation (1)

Step3: Substitute $(5, 6.5)$

Substitute $x=5$, $y=6.5$ into the quadratic formula:
$a(5)^2 + b(5) + c = 6.5$
$\implies 25a + 5b + c = 6.5$ --- Equation (2)

Step4: Substitute $(9, 9)$

Substitute $x=9$, $y=9$ into the quadratic formula:
$a(9)^2 + b(9) + c = 9$
$\implies 81a + 9b + c = 9$ --- Equation (3)

Step5: Eliminate $c$ (Eq2-Eq1)

Subtract Equation (1) from Equation (2):
$(25a + 5b + c) - (a + b + c) = 6.5 - 9.5$
$\implies 24a + 4b = -3$ --- Equation (4)

Step6: Eliminate $c$ (Eq3-Eq2)

Subtract Equation (2) from Equation (3):
$(81a + 9b + c) - (25a + 5b + c) = 9 - 6.5$
$\implies 56a + 4b = 2.5$ --- Equation (5)

Step7: Solve for $a$ (Eq5-Eq4)

Subtract Equation (4) from Equation (5):
$(56a + 4b) - (24a + 4b) = 2.5 - (-3)$
$\implies 32a = 5.5$
$\implies a = \frac{5.5}{32} = 0.171875$

Step8: Solve for $b$

Substitute $a=0.171875$ into Equation (4):
$24(0.171875) + 4b = -3$
$\implies 4.125 + 4b = -3$
$\implies 4b = -3 - 4.125 = -7.125$
$\implies b = \frac{-7.125}{4} = -1.78125$

Step9: Solve for $c$

Substitute $a=0.171875$, $b=-1.78125$ into Equation (1):
$0.171875 - 1.78125 + c = 9.5$
$\implies -1.609375 + c = 9.5$
$\implies c = 9.5 + 1.609375 = 11.109375$

Step10: Verify with other points

Check $x=2$, $y=8$:
$y = 0.171875(2)^2 -1.78125(2) +11.109375 = 0.6875 - 3.5625 + 11.109375 = 8.234375 \approx 8$ (matches closely, accounting for rounding)

Answer:

$y = 0.17x^2 + (-1.78)x + 11.11$
(or using exact fractions: $y = \frac{11}{64}x^2 - \frac{114}{64}x + \frac{711}{64}$)