/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 33 The following table shows U.S. f... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 33

The following table shows U.S. first-class stamp prices (per ounce) over time. $$ \begin{array}{lc} \hline \text { Year } & \text { Price for First-Class Stamp } \\ \hline 2001 & 34 \text { cents } \\ 2002 & 37 \text { cents } \\ 2006 & 39 \text { cents } \\ \hline \end{array} $$ a. Construct a step function describing stamp prices for \(2001-2006 .\) b. Graph the function. Be sure to specify whether each of the endpoints is included or excluded. c. In 2007 the price of a first-class stamp was raised to 41 cents. How would the function domain and the graph change?

Short Answer

Expert verified
Step function: \[ P(x) = \begin{cases} 34 & \text{if } 2001 \leq x < 2002 \ 37 & \text{if } 2002 \leq x < 2006 \ 39 & \text{if } 2006 \leq x < 2007 \ 41 & \text{if } 2007 \leq x < 2008 \end{cases} \]Graph: Plot with appropriate intervals and closed/open circles.

Step by step solution

01

- Identify price intervals

Determine the intervals for each year. For this table, intervals are as follows: 2001, 2002-2005, 2006.
02

- Construct the step function

Define the step function, indicating price adjustments for each interval:\[ P(x) = \begin{cases} 34 & \text{if } 2001 \leq x < 2002 \ 37 & \text{if } 2002 \leq x < 2006 \ 39 & \text{if } 2006 \leq x < 2007 \end{cases} \]
03

- Graph the function

Plot the step function on a graph with the x-axis representing years and the y-axis representing price in cents. Include endpoints as follows:\[ \text{Closed (solid) circles at } (2001, 34), (2002, 37), \text{ and } (2006, 39) \]\[ \text{Open (hollow) circles at } (2002, 34) \text{ and } (2006, 37) \]
04

- Extend the function to 2007

Add the interval for 2007 and adjust the domain:\[ P(x) = \begin{cases} 34 & \text{if } 2001 \leq x < 2002 \ 37 & \text{if } 2002 \leq x < 2006 \ 39 & \text{if } 2006 \leq x < 2007 \ 41 & \text{if } 2007 \leq x < 2008 \end{cases} \]
05

- Update the graph

Extend the graph to include 2007 with closed circles at (2007, 41) and open circles at (2008, 41). The new graph displays the updated prices accordingly.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Functions
Graphing functions is like creating a visual story of how values change. When we graph the step function describing stamp prices from 2001 to 2006, we plot the years on the x-axis and the price in cents on the y-axis. Start by plotting the given points. For instance, in 2001, the price is 34 cents, so plot a point at (2001, 34). Each interval on the graph will look like a horizontal line, reflecting that the price remains constant until it changes again. Remember to use closed (solid) circles to denote included endpoints and open (hollow) circles to mark excluded endpoints. This visual storytelling helps us better understand the changes over time.
Function Intervals
Function intervals tell us which input (x) values apply to each part of a function. For the stamp prices, the intervals break down as follows: 2001, 2002-2005, and 2006. Each interval shows a period during which the price remains stable before it jumps to a new level. It's crucial to correctly identify and label these intervals because they dictate how our step function behaves. For example, the interval \[2002, 2006\] tells us that from the start of 2002 to the end of 2005 the price was 37 cents. This organization makes it easier to understand the different segments of the function.
Piecewise Functions
Piecewise functions involve multiple sub-functions, each defined over a specific interval. The stamp price problem is a perfect example. We define our step function piece by piece: \[ P(x) = \begin{cases} 34 & \text{if } 2001 \leq x < 2002 \ 37 & \text{if } 2002 \leq x < 2006 \ 39 & \text{if } 2006 \leq x < 2007 \end{cases} \] Here, each 'case' of P(x) corresponds to a different price for a different time period. This form allows us to represent complex scenarios where a single formula isn't enough. Each piece of the piecewise function has a clear role in the overall function.
Data Representation in Mathematics
Representing data mathematically can turn raw numbers into meaningful information. In this exercise, we start with a simple table of stamp prices over time, then translate those numbers into a step function. This function and its graph help us visualize the history of stamp prices easily. Bullet points are another helpful tool for data representation, like listing the price in each interval:
鈼� 34 cents in 2001
鈼� 37 cents from 2002 to 2005
鈼� 39 cents in 2006
By converting a list of values into graphical and functional forms, we make it much easier to see patterns, trends, and changes. These methods help us analyze and understand data in more depth.

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Most popular questions from this chapter

Calculate the average rate of change between adjacent points for the following function. (The first few are done for you.) $$ \begin{array}{ccc} \hline & & \text { Average Rate } \\ x & f(x) & \text { of Change } \\ \hline 0 & 0 & \text { n.a. } \\ 1 & 1 & 1 \\ 2 & 8 & 7 \\ 3 & 27 & \\ 4 & 64 & \\ 5 & 125 & \\ \hline \end{array} $$ a. Is the function \(f(x)\) increasing, decreasing, or constant throughout? b. Is the average rate of change increasing, decreasing, or constant throughout?

Calculate the average rate of change between adjacent points for the following functions and place the values in a third column in each table. (The first entry is "n.a.") $$ \begin{array}{rr} \hline x & f(x) \\ \hline 0 & 5 \\ 10 & 25 \\ 20 & 45 \\ 30 & 65 \\ 40 & 85 \\ 50 & 105 \\ \hline \end{array} $$ $$ \begin{array}{rl} \hline x & g(x) \\ \hline 0 & 270 \\ 10 & 240 \\ 20 & 210 \\ 30 & 180 \\ 40 & 150 \\ 50 & 120 \\ \hline \end{array} $$ a. Are the functions \(f(x)\) and \(g(x)\) increasing, decreasing, or constant throughout? b. Is the average rate of change of each function increasing, decreasing, or constant throughout?

For each of the given points write equations for three lines that all pass through the point such that one of the three lines is horizontal, one is vertical, and one has slope \(2 .\) a. (1,-4) b. (2,0)

(Graphing program optional.) The accompanying table indicates the number of juvenile arrests (in thousands) in the United States for aggravated assault. $$ \begin{array}{ccc} \hline & \begin{array}{c} \text { Juvenile } \\ \text { Arrests } \\ \text { Year } \end{array} & \begin{array}{c} \text { Annual Average } \\ \text { (thousands) } \end{array} & \begin{array}{c} \text { Rate of Change } \\ \text { over Prior 5 Years } \end{array} \\ \hline 1985 & 36.8 & \text { n.a. } \\ 1990 & 54.5 & \\ 1995 & 68.5 & \\ 2000 & 49.8 & \\ 2005 & 36.9 & \\ \hline \end{array} $$ a. Fill in the third column in the table by calculating the annual average rate of change. b. Graph the annual average rate of change versus time. c. During what 5 -year period was the annual average rate of change the largest? d. Describe the change in aggravated assault cases during these years by referring both to the number and to the annual average rate of change.

(Graphing program optional.) Solve each equation for \(y\) in terms of \(x\), then identify the slope and the \(y\) -intercept. Graph each line by hand. Verify your answers with a graphing utility if available. a. \(-4 y-x-8=0\) b. \(\frac{1}{2} x-\frac{1}{4} y=3\) c. \(-4 x-3 y=9\) d. \(6 x-5 y=15\)
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