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Visual or Graphical Representations
The equation y = 3x + 2, where x represents the first number and y represents the second number. We could also write this as s = 2f + 2, where f represents the first number and s represents the second number.
The same relationship holds whether we represent it verbally, numerically, visually, or symbolically. What's more, there are important characteristics of linear relationships that are revealed in each type of representation.
A table of values or a graph may best reveal one characteristic of all linear relationships, that of constant growth:
- For each unit change in the first variable, there is a corresponding constant change in the second variable.
In the relationship illustrated above, the constant change is 3 units. For each change of 1 unit in the first variable, the second variable changes 3 units. This essential characteristic of linear relationships is called slope. The origin of that term is best related to the visual representation, where we can think of slope relating to the inclination of the line:
- For every 1-unit change horizontally, there is a 3-unit change vertically.
If we know two ordered pairs (x1,y1) and (x2,y2) that are part of a linear relationship, we have enough information to determine the slope of the relationship.
The first line shown above represents the concept of slope: constant change. The second line shows how to calculate the slope when we know two ordered pairs in the relationship. The last symbol in the second line shows the traditional symbol used to represent slope, the letter m.
Other important aspects of linear relationships include the location of the axes intercepts. That is, we typically identify the ordered pairs that describe where the line intersects the x-axis and where it intersects the y-axis. In the form of ordered pairs, that means we want to know the values a and b in the ordered pairs (a,0) and (0,b). The first is the x-axis intercept and the second the y-axis intercept. When real-life situations are described through linear relationships, the intercepts often take on important meaning. The following example is intended to illustrate that.
Linear Relatonships: An Example
- At Captain's Video Emporium, video club members rent Nintendo games for $2 per night. The club also charges a $20 annual membership fee.
Here is a table of values to show the cost associated with renting various numbers of Nintendo games. The cost includes the annual fee plus the per-night rental charge.
We can also plot the ordered pairs (number of rentals,total cost), as shown in the graph below.
In symbolic form, we can write an equation to represent the relationship between games rented and total cost. If we let g represent the number of games rented and T represent the total cost, then T = 2g + 20 represents the relationship.
We have shown three different ways to represent the relationship: numerical, graphical, and symbolic. The fourth representation, verbal, was the original description of the relationship.
How does the graph below differ from the one above? How are they similar? What is the significance of the vertical-axis intercept? Why does only one end of the line segment have arrowheads on it?
These and related questions are important to consider when working with representations of relationships. Here are some questions and answers that help focus on critical aspects of the relationship.
- Can a club member be charged less than $20? Why or why not? How is this shown in the graph?
- Can a club member rent fewer than zero (0) games? Why or
why not? How is this shown in the graph?
In the real world, no member can rent fewer than zero games, and even if a member rents no games, the person still would pay the $20 membership fee. The graph shows this because it stops at the ordered pair (0,20).
- Can anyone rent some fractional part of a video game, such as 3 1/4 games? Why or why not? How is this shown in the graph?
- Can someone spend a total of $143 on video games at Captain
Video's? Explain.
In the real world, a member cannot rent a part of a game, because we assume each game is a physical whole that cannot be split into parts. The graph is deceiving in this way, for the solid line seems to indicate that any value on the horizontal axis is possible. To be more accurate, we could try to draw in many many little dots to indicate only certain values are possible. However, this is not practical. So we draw in the solid line and then somehow indicate that only whole values are possible.
The same is true for the vertical axis. No member can spend exactly $143 on game rentals, because after paying the $20 membership fee, there would be $123 left for rentals. Each game, however, costs $2 to rent, and $123 is not a multiple of $2, so we could not rent a whole number of games with exactly that amount of money.
- Is there a limit to the number of games that can be rented?
How is this shown in the graph?
We know of no restrictions on the number of games that could be rented. The segment in the graph shows arrowheads at one end, indicating that the pattern shown to that point continues. In practical terms, there may be some limit to the number of games that can be rented, most likely based on the inventory maintained by Captain Video. Although we can mention that as a possibility, we have no specific evidence that verifies that. Thus, we need the arrowheads to show the pattern can go on and on, just as we cannot have arrowheads on the other end of the segment, because we want to show there is a restriction in the pattern.
- What is the cost per rental? How is this shown in the
graph? How is it shown in the equation? How is it shown in the
table?
The cost per rental is another way of describing the rate of change in the total cost relative to the number of video games rented. This is a constant value, $2 per game. This is the slope of the line shown in the graph, and is found by determining how much change in total cost (vertical change) there is for each unit change in number of rentals (horizontal change). In the equation T = 2g + 20, the slope is the coefficient of the horizontal-axis variable. This is true for all linear equations written in this form.
In the table of values, we can see that for equal changes in the number of games rented, there is a constant change in the total cost. For example, increasing the number of games rented from 5 to 10 results in a $10 increase in total costs. For any 5-game increase in the number of games rented, the total cost increases by $10. This constant rate of change is a fundamental characteristic of linear relationships.
As we wrap up this brief review of linear relationships, consider the four situations described below. For each case:
- Identify the two variables to be related.
- Represent the relationship in at least two other ways (graphically, numerically, or symbolically).
- Determine whether the relationship is linear. If it is not, explain why not. If it is linear, determine the constant rate of change.
Case I: The treadmill was set to revolve at 4 miles per hour.
Case II:
Case III: The owner of the ice cream stand determined that each single-dip cone costs 36 cents to produce, and that the portable sales stand required $200 to open each weekend.
Case IV:
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