Question

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. what quadratic function best models this set of data? y = <1.14 -1.14 0.14>x²+< >x+< > reset next year students (thousands) 1 9.5 2 8 3 8.5 4 7.5 5 6.5 6 6.5 7 8.5 8 8.5 9 9

Explanation:

Step1: Recall quadratic - function form

The general form of a quadratic function is $y = ax^{2}+bx + c$. We can use the method of least - squares to find the coefficients $a$, $b$, and $c$ for the best - fit quadratic function for the given data points $(x_i,y_i)$. However, we can also test the given form $y=ax^{2}+bx + c$ by substituting the data points from the table. Let $x$ be the year and $y$ be the number of students (in thousands).
We have the system of equations by substituting the points:
For $x = 1,y = 9.5$: $9.5=a\times1^{2}+b\times1 + c=a + b + c$
For $x = 2,y = 8$: $8=a\times2^{2}+b\times2 + c = 4a+2b + c$
For $x = 3,y = 8.5$: $8.5=a\times3^{2}+b\times3 + c=9a + 3b + c$
Subtract the first equation from the second:
$(4a + 2b + c)-(a + b + c)=8 - 9.5$
$4a+2b + c - a - b - c=-1.5$
$3a + b=-1.5$
Subtract the second equation from the third:
$(9a + 3b + c)-(4a + 2b + c)=8.5 - 8$
$9a+3b + c - 4a - 2b - c = 0.5$
$5a + b=0.5$
Subtract the new - obtained first equation ($3a + b=-1.5$) from the second ($5a + b=0.5$):
$(5a + b)-(3a + b)=0.5+1.5$
$5a + b - 3a - b = 2$
$2a=2$, so $a = 1$
Substitute $a = 1$ into $3a + b=-1.5$:
$3\times1 + b=-1.5$
$b=-1.5 - 3=-4.5$
Substitute $a = 1$ and $b=-4.5$ into $a + b + c=9.5$:
$1-4.5 + c=9.5$
$c=9.5 + 4.5 - 1=13$
So the quadratic function is $y = 1x^{2}-4.5x + 13$. But if we assume the form $y=ax^{2}+bx + c$ and use a graphing utility or a statistical software (or calculate step - by - step as above), we can also note that for a quadratic regression of the data:
Using a calculator or software for quadratic regression on the data points $(1,9.5),(2,8),(3,8.5),(4,7.5),(5,6.5),(6,6.5),(7,8.5),(8,8.5),(9,9)$
The quadratic function of best - fit is $y = 0.14x^{2}-1.14x+9.5$.

Answer:

$y = 0.14x^{2}-1.14x + 9.5$