The following table shows the percentage of the U.S....

The following table shows the percentage of the U.S. Discretionary Budget allocated to education in 2003,2005, and 2009 . (See Exercise $20 ; t=3$ represents the start of $2003 .)^{10}$ $$ \begin{array}{|r|c|c|c|} \hline \text { Year } \boldsymbol{t} & 3 & 5 & 9 \\ \hline \text { Percentage } & 6.8 & 7 & 6.2 \\ \hline \end{array} $$ Find a quadratic model for these data, and use your model to estimate the percentage of the U.S. Discretionary Budget allocated to education in 2008 . Compare your answer with the actual figure shown in Exercise 20 .

The following table shows the percentage of the U.S. Discretionary Budget allocated to education in 2003,2005, and 2009 . (See Exercise $20 ; t=3$ represents the start of $2003 .)^{10}$
$$
\begin{array}{|r|c|c|c|}
\hline \text { Year } \boldsymbol{t} & 3 & 5 & 9 \\
\hline \text { Percentage } & 6.8 & 7 & 6.2 \\
\hline
\end{array}
$$
Find a quadratic model for these data, and use your model to estimate the percentage of the U.S. Discretionary Budget allocated to education in 2008 . Compare your answer with the actual figure shown in Exercise 20 .

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Key Concepts

Systems of Equations

Determining the coefficients of a quadratic model from data involves setting up a system of equations. Each data point provides an equation when substituted into the quadratic form, and solving this system yields the unknown coefficients that define the model.

Estimation and Prediction

Once a quadratic model is established, it can be used to estimate or predict values for inputs that were not part of the original data set. This concept, often referred to as interpolation or, in cases outside the original data range, extrapolation, is key in analyzing trends and making future forecasts based on past data.

Quadratic Functions

A quadratic function is a second-degree polynomial expressed in the form ax² + bx + c. It is fundamental in modeling parabolic trends and behaviors, and its graphical representation, a parabola, can open upward or downward depending on the sign of the coefficient a.

Polynomial Interpolation

Polynomial interpolation is the process of constructing a polynomial that exactly passes through a set of given data points. In quadratic interpolation, a second-degree polynomial is determined to fit three data points, providing a smooth model that can be used to estimate values within the range of the data.

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