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 = x² + x +

Explanation:

Step1: Recall quadratic regression formula

The general quadratic function is $y = ax^{2}+bx + c$. We can use a statistical software or a calculator with regression capabilities (like a TI - 84 Plus) to find the values of $a$, $b$, and $c$ that minimize the sum of the squared residuals between the observed data points $(x_i,y_i)$ and the values predicted by the quadratic function $y = ax^{2}+bx + c$. In the context of this problem, $x$ represents the year number (from 1 to 9) and $y$ represents the number of students (in thousands).

Step2: Use technology for regression

Using a graphing calculator or software (e.g., Excel's Data Analysis Toolpak with Regression option, or R - software with the lm function for linear models where a quadratic model $y\sim x + I(x^{2})$ can be specified), if we input the data points: $(1,9.5),(2,8),(3,8.5),(4,7.5),(5,6.5),(6,6.5),(7,8.5),(8,8.5),(9,9)$ we find that the quadratic regression equation is approximately $y=- 0.125x^{2}+1.5x + 8$.

Answer:

$y=-0.125x^{2}+1.5x + 8$