This paper is an example of an introductory level algebra homework assignment. It describes a Cartesian graph where x = y and provides an example of that graph. It discusses the equations for horizontal and vertical lines. A horizontal line is represented by the equation y= some number and a vertical line is represented by the equation x= some number. It defines the following terms: expression, equation, and function.
¶ … Cartesian graph is approximately the distance y in feet a person walks in x hours. What does the slope of this line represent? How is this graph useful? Provide another example for your colleagues to explain.
If a line on a Cartesian graph is approximately the distance y in feet that a person walks in x hours, then there would be a 1:1 correspondence between the x and y coordinates, so that the equation for the line would be y=x, and the slope of the line would be 1. The equation for the line can be seen below:
(Harmsworth, 2012).
The graph is useful because it helps demonstrate the one to one correspondence between the two variables, and makes it clear that for each additional hour (x) spent walking, the person travels an additional set distance in feet (y). However, it is unlikely that this equation would actually represent a person's distance in feet over a certain number of hours. Expanding the graph, one would expect to see a decline in feet per hour as exhaustion set in for the walker. As a result, it is important to realize that this standard graph has some limitations. Another example of this type of one to one graphical equation would be that for every text message Johnny receives (x), he sends a text message in reply (y).
If a line has no y-intercept what can you say about the line?
If a line has no y-intercept, then there is no value of y for which the value of x would be zero. This line is going to be a vertical line and must run parallel to the y-axis; because, otherwise the equation would include a value at which x would be zero. The form of the equation for such a line will be x= a number. In the example below, the equation would be x=5.
(Rehill, 2012).
What if a line has no x-intercept?
If a line has no x-intercept, then there is no value of x for which the y would be zero. This line is going to be a horizontal line and must run parallel to the x-axis; because, otherwise the equation would include a value at which y would be zero. The form of the equation for such a line will be y= a number. In the example below, the equation would be y=5.
(Rehill, 2012).
Think of a real life situation where a graph would have no x- or y-intercept. Will what you say about the line always be true in that situation?
A situation where a graph would have no x- or y-intercept would be one where the payment for something is constant, for example, an all-you can eat buffet where the fixed cost is $5. If x is the number of plates of food eaten, and y is the price, the table of coordinates might be as follows:
Plates (x)
Price (y)
1
5
2
5
3
5
4
5
5
5
The equation for that would be represented by y=5 and the graph of the equation would be the same as in the sample graph for y=5 provided above.
Interestingly, in real life, what one would say about the line would probably only be true for a fixed set of x values. For example, it is impossible to eat a negative number of plates of food. Likewise, even all-you can eat buffets generally limit their diners to one meal-period, so that there is a finite number of plates the diner could consume. However, those conditions would actually change the form of the equation, transform it from a simple linear equation, and would not be represented by the conditions described.
What are the differences among expressions, equations, and functions? Provide examples of each.
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