Question

Find the rate of change between the two points. Give the units of measure for the rate. \((3,5)\) and \((11,69) ; x\) in years, \(y\) in dollars.

Short answer

The rate of change between the two points is 8 dollars per year.

Step-by-step solution

Understand what the coordinate points represent

The coordinates \((3,5)\) and \((11,69)\) each represent two points. Where the first value of the pair is 'x' representing the years, and the second value of the pair is 'y' indicating dollars. So, \(3\) years corresponds to \$5 whereas \(11\) years corresponds to \$69.

Compute the difference in 'y' values

The 'y' value difference or 'rise' is obtained by subtracting the 'y' coordinates of the two points. In this case, it's \(69-5 = 64\) dollars.

Compute the difference in 'x' values

The 'x' value difference or 'run' is obtained by subtracting the 'x' coordinates of the two points. Here, it's \(11-3 = 8\) years.

Calculate the rate of change or slope

The rate of change or slope is the ratio of the difference in 'y' values to the difference in 'x' values. Thus, the rate of change is \(\frac{64}{8} = 8\) dollars per year.

Key concepts

Coordinate Points

Understanding coordinate points is fundamental in graphically representing relationships between two variables. In our context, each coordinate point, such as (3,5), consists of an 'x' value and a 'y' value. These are plotted on a two-dimensional grid where the 'x' value corresponds to the horizontal position (often time or another independent variable), and the 'y' value represents the vertical position (typically a dependent variable, such as money in this exercise).
Let's look at the example provided. The coordinate pairs (3,5) and (11,69) symbolize two specific instances in time – 3 years and 11 years respectively. The 'y' values associated with these points, 5 and 69 dollars, tell us about the amount of money at those instances. By examining these coordinates, we can trace the journey from one point in time to another, observing the changes that occur in the dependent variable, in this case, the amount in dollars.

Slope Calculation

When considering the concept of slope, think of it as a measure of steepness or inclination. In the context of a graph, it describes how much the 'y' variable changes for a unit increase in the 'x' variable. To calculate the slope between two points, you need to understand the concept of 'rise over run'. This essentially means you divide the change in the 'y' values by the change in the 'x' values between two points.
For our exercise, the slope is determined by first computing the rise, which is the difference in dollars (69-5 = 64 dollars) and then calculating the run, which is the difference in years (11-3 = 8 years). The slope, or the rate of change, is the ratio between these two differences and represents the change in dollars per year. Performing this calculation \(\frac{64}{8} = 8\), we find that the rate of change is 8 dollars per year. This understanding allows us to predict financial growth over time or to compare rates of change between different scenarios.

Linear Functions

A linear function is one of the simplest forms of a mathematical function and is identified by a straight-line graph. It has the general form \(y = mx + b\), where 'm' denotes the slope and 'b' represents the y-intercept—the 'y' value when 'x' is zero. The slope 'm' tells us how much 'y' changes for a one-unit increase in 'x'.
In a linear relationship, as seen in our initial exercise, the change between any two points on the line will always have the same rate, or in other words, the slope remains constant. This constant rate of change is what makes the graph straight. In real-world terms, it implies a consistent growth pattern. For every additional year, the amount of money increases by the same rate, 8 dollars per year. Learning to work with linear functions is not only essential for algebra but also has practical applications in fields like economics, social sciences, and natural sciences.