Poverty vs. Income Based on data from 1988 through 2003 ,...

Poverty vs. Income Based on data from 1988 through 2003 , the poverty rate (percentage of households with incomes below the poverty threshold) in the U.S. can be approximated by $p(x)=0.092 x^{2}-8.1 x+190$ percentage points $(38 \leq x \leq 44)$ where $x$ is the U.S. median household income in thousands of dollars. ${ }^{43}$ a. Use technology to complete the following table which shows the average rate of change of $p$ over successive intervals of length $\frac{1}{2} .$ (Round all answers to two decimal places.) HINT [See Example 4.] $$ \begin{array}{|r|r|r|r|r||r|r|} \hline \text { Interval } & {[39,39.5]} & {[39.5,40]} & {[40,40.5]} & {[40.5,41]} & {[41,41.5]} & {[41.5,42]} \\ \hline \begin{array}{r} \text { Average } \\ \text { Rate of } \\ \text { Change of } p \end{array} & & & & & & \\ \hline \end{array} $$ b. Interpret your answer for the interval $[40,40.5]$, being sure to indicate the direction of change and the units of measurement. c. Multiple choice: As the median household income rises, the poverty rate (A) increases (B) decreases (C) increases, then decreases (D) decreases, then increases d. Multiple choice: As the median income increases, the effect on the poverty rate is (A) more pronounced (B) less pronounced

Poverty vs. Income Based on data from 1988 through 2003 , the poverty rate (percentage of households with incomes below the poverty threshold) in the U.S. can be approximated by $p(x)=0.092 x^{2}-8.1 x+190$ percentage points $(38 \leq x \leq 44)$
where $x$ is the U.S. median household income in thousands of dollars. ${ }^{43}$
a. Use technology to complete the following table which shows the average rate of change of $p$ over successive intervals of length $\frac{1}{2} .$ (Round all answers to two decimal places.) HINT [See Example 4.]
$$
\begin{array}{|r|r|r|r|r||r|r|}
\hline \text { Interval } & {[39,39.5]} & {[39.5,40]} & {[40,40.5]} & {[40.5,41]} & {[41,41.5]} & {[41.5,42]} \\
\hline \begin{array}{r}
\text { Average } \\
\text { Rate of } \\
\text { Change of } p
\end{array} & & & & & & \\
\hline
\end{array}
$$
b. Interpret your answer for the interval $[40,40.5]$, being sure to indicate the direction of change and the units of measurement.
c. Multiple choice: As the median household income rises, the poverty rate
(A) increases
(B) decreases
(C) increases, then decreases
(D) decreases, then increases
d. Multiple choice: As the median income increases, the effect on the poverty rate is
(A) more pronounced
(B) less pronounced

Key Concepts

Average Rate of Change

The average rate of change of a function over an interval is defined as the change in the function’s value divided by the change in the independent variable. This concept is equivalent to finding the slope of the secant line that connects two points on the function’s graph and provides an approximation of the behavior of the function over that interval.

Interpreting Rates in Context

Interpreting the average rate of change in an applied context involves relating the mathematical rate to a real-world situation. It requires an understanding of the units involved and assessing whether the function is increasing or decreasing over a given interval, thereby helping to explain the underlying relationship between the variables being studied.

Quadratic Functions

A quadratic function, characterized by a second-degree polynomial, produces a parabolic graph that can model various real-world phenomena. Understanding the properties of quadratic functions, such as their vertex, axis of symmetry, and the fact that their rate of change is not constant, is essential when analyzing how changes in the input affect the output.

Technology-Assisted Analysis

Using technology to analyze functions involves utilizing graphing calculators or software to compute function values, estimate average rates of change, and visualize data. This approach enhances understanding by providing precise numerical results and clear graphical representations of how a function behaves over specific intervals.

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