Problem 83 - Algebra A Combined Function

Question

There were approximately 2025 heart transplants performed in the United States in \(2004 .\) In \(2007,\) the number of heart transplants in the United States rose to 2208. (Source: Organ Procurement and Transplantation Network) a. Write two ordered pairs of the form (year, number of heart transplants). b. Find the slope of the line between the two points. c. Write a sentence explaining the meaning of the slope as a rate of change.

Step-by-Step Solution

Answer

Ordered pairs: (2004, 2025), (2007, 2208). Slope: 61. The slope indicates an average increase of 61 transplants per year between 2004 and 2007.

Step 1: Identify the Ordered Pairs

The problem gives us information for two years, 2004 and 2007. For 2004, the number of transplants is 2025, and for 2007, it's 2208. Thus, the ordered pairs representing these data points are (2004, 2025) and (2007, 2208).

Step 2: Calculate the Slope

To find the slope, we use the formula for the slope of a line passing through two points, \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the values from our ordered pairs, we have \( m = \frac{2208 - 2025}{2007 - 2004} = \frac{183}{3} = 61 \).

Step 3: Interpret the Slope

The slope represents the rate of change in the number of heart transplants per year. A slope of 61 means that each year, between 2004 and 2007, there was an average increase of 61 heart transplants performed annually.

Key Concepts

Ordered Pairs

Ordered pairs are a fundamental concept in algebra and coordinate geometry. They help us to understand and represent relationships between two variables on a graph. In this exercise, we consider the years and the corresponding number of heart transplants as our variables.

An ordered pair is typically written as \((x, y)\), where \(x\) represents the first component, and \(y\) represents the second. In our heart transplant exercise:

• The first ordered pair is \((2004, 2025)\), where 2004 is the year, and 2025 is the number of heart transplants.
• The second ordered pair is \((2007, 2208)\), where 2007 is the year, and 2208 is the number of heart transplants.

These ordered pairs allow us to plot the information on a coordinate plane and visualize changes over time.

Slope Calculation

The slope of a line is a measure of its steepness and direction. It's a crucial part of linear equations in algebra because it tells us how quickly something changes. We can calculate the slope between two points using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here's how you can think about it:

• The numerator, \(y_2 - y_1\), represents the change in the \(y\)-values (what you're measuring, here the number of heart transplants).
• The denominator, \(x_2 - x_1\), represents the change in the \(x\)-values (the time elapsed, in this case, in years).

Using the ordered pairs from the exercise:

• Take the difference in the number of transplants: \(2208 - 2025 = 183\).
• Take the difference in years: \(2007 - 2004 = 3\).
• The slope \( m \) is \( \frac{183}{3} = 61 \).

The slope tells us the number of additional heart transplants performed each year during this period.

Rate of Change

The term "rate of change" in mathematics often refers to how one quantity changes in relation to another. It's closely linked to the concept of slope. When you calculate the slope of a line, you are essentially finding the rate at which one variable changes compared to another.

In our heart transplant scenario, the slope of 61 has a special significance. It represents the rate of change in the number of heart transplants over time:

• Every year between 2004 and 2007, the number of heart transplants increased by an average of 61.
• This gives a clear, easy-to-understand measure of how rapidly the number is changing.

Understanding rate of change is vital in many areas like physics, economics, and statistics, giving insights into patterns and trends over time.