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

The following chart shows the percentage of the U.S. Discretionary Budget allocated to education from 2003 to 2009 ( $t=3$ represents the start of 2003 ). a. If you want to model the percentage figures with a function of the form $$ f(t)=a t^{2}+b t+c $$ would you expect the coefficient $a$ to be positive or negative? Why? HINT [See "Features of a Parabola" page 621.] b. Which of the following models best approximates the data given? (Try to answer this without actually computing values) (A) $f(t)=0.04 t^{2}+0.3 t-6$ (B) $f(t)=-0.04 t^{2}+0.3 t+6$ (C) $f(t)=0.04 t^{2}+0.3 t+6$ (D) $f(t)=-0.04 t^{2}+0.3 t-6$ c. What is the nearest year that would correspond to the vertex of the graph of the correct model from part (b)? What is the danger of extrapolating the data in either direction?

The following chart shows the percentage of the U.S. Discretionary Budget allocated to education from 2003 to 2009 ( $t=3$ represents the start of 2003 ).
a. If you want to model the percentage figures with a function of the form
$$
f(t)=a t^{2}+b t+c
$$
would you expect the coefficient $a$ to be positive or negative? Why? HINT [See "Features of a Parabola" page 621.]
b. Which of the following models best approximates the data given? (Try to answer this without actually computing values)
(A) $f(t)=0.04 t^{2}+0.3 t-6$
(B) $f(t)=-0.04 t^{2}+0.3 t+6$
(C) $f(t)=0.04 t^{2}+0.3 t+6$
(D) $f(t)=-0.04 t^{2}+0.3 t-6$
c. What is the nearest year that would correspond to the vertex of the graph of the correct model from part (b)? What is the danger of extrapolating the data in either direction?

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Key Concepts

Extrapolation

Extrapolation involves using a model to predict values outside the range of the observed data. This practice can be dangerous because the behavior of the trend outside the established range is uncertain and may not follow the same pattern, leading to unreliable or misleading conclusions.

Vertex of a Parabola

The vertex of a parabola is the point where the function reaches its maximum or minimum value, depending on whether the parabola opens downward or upward, respectively. The vertex's t-coordinate can be calculated as -b/(2a), which provides the point in time corresponding to the peak or trough of the modeled data.

Quadratic Functions

Quadratic functions are second-degree polynomials represented in the form f(t) = at² + bt + c. They are used to model various types of data, especially when the change in the data is non-linear, leading to a parabolic graph which can indicate acceleration or deceleration in the trend.

Parabola Orientation

The orientation of the parabola, or the direction in which it opens, is determined by the sign of the leading coefficient 'a'. A positive 'a' means the parabola opens upward (indicative of a minimum vertex), while a negative 'a' means it opens downward (indicative of a maximum vertex). This orientation is crucial in selecting the appropriate model for the data trend.

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