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course PHY 231
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Describing Graphs*********************************************
Question: `q001. You will frequently need to describe the graphs you have constructed in this course. This exercise is designed to get you used to some of the terminology we use to describe graphs. Please complete this exercise and submit your work as instructed.
Note that you should do these graphs on paper without using a calculator. None of the arithmetic involved here should require a calculator, and you should not require the graphing capabilities of your calculator to answer these questions.
Problem 1. We make a table for y = 2x + 7 as follows: We construct two columns, and label the first column 'x' and the second 'y'. Put the numbers -3, -2, -1, -, 1, 2, 3 in the 'x' column.We substitute -3 into the expression and get y = 2(-3) + 7 = 1. We substitute -2 and get y = 2(-2) + 7 = 3. Substituting the remaining numbers we get y values 5, 7, 9, 11 and 13. These numbers go into the second column, each next to the x value from which it was obtained. We then graph these points on a set of x-y coordinate axes. Noting that these points lie on a straight line, we then construct the line through the points.
Now make a table for and graph the function y = 3x - 4.
Identify the intercepts of the graph, i.e., the points where the graph goes through the x and the y axes.
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Your solution:
The graph goes through the x axis when y = 0 and through the y axis when x = 0.
The x-intercept is x = 4/3.
The y-intercept is when y = 3 * 0 - 4 = -4 and the x intercept is at (4/3, 0) and the y intercept is at (0, -4).
confidence rating #$&*: 3
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Self-critique (if necessary): no
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Self-critique Rating: 3
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Question: `q002. Does the steepness of the graph in the preceding exercise (of the function y = 3x - 4) change? If so describe how it changes.
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Your solution:
When I graphed it I did not see a change in steepness
confidence rating #$&*: 3
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Self-critique (if necessary): no
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Self-critique Rating: 3
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Question: `q003. What is the slope of the graph of the preceding two exercises (the function is y = 3x - 4;slope is rise / run between two points of the graph)?
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Your solution:
The formula to do this is ((x2-x1)/(y2-y1))
Hence using the data from previous problem it is y = 3 * 2 - 4 = 2 and when x = 8 we have y = (3 * 8) - 4 = 20. Then you minus it for rise 20 - 2 = 18 and the run is 8 - 2 = 6 so the slope is rise / run = 18 / 6 = 3.
confidence rating #$&*:: 3
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Self-critique (if necessary): no
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Self-critique Rating: 2
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Question: `q004. Make a table of y vs. x for y = x^2. Graph y = x^2 between x = 0 and x = 3.
Would you say that the graph is increasing or decreasing?
Does the steepness of the graph change and if so, how?
Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?
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Your solution:
Graph points include (0,0), (1,1), (2,4) and (3,9). The y values are 0, 1, 4 and 9,
The increases between these points are 1, 3 and 5, so the graph not only increases, but it proportionally increases at an increasing rate
confidence rating #$&*:: 3
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Self-critique (if necessary): Needs better wording
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Self-critique Rating: 2
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Question: `q005. Make a table of y vs. x for y = x^2. Graph y = x^2 between x = -3 and x = 0.
Would you say that the graph is increasing or decreasing?
Does the steepness of the graph change and if so, how?
Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?
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Your solution:
I used a calculator to graph these functions. As we moved from left to right the y values were 9, 4, 1, 0 in decreasing order for (-3,9), (-2,4), (-1,1), (0,0). Though you can see that the magnitudes of the changes, we test this in math and see that the x from 9 goes to 4 and then to 1 to 0 in decreasing order, so the steepness is decreasing.
Thus the interval is in decreasing rate.
confidence rating #$&*:: 3
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A calculator is not appropriate for graphing these basic functions. You need to know how the graph is constructed and think about why it has the shape it does, in the context of the arithmetic of evaluating the function.
Graphs are to be constructed in this manner, and more complex graphs by using the methods of standard precalculus (stretching and shifting in the horizontal and vertical directions).
To answer the question of the description of the graph, you get a completely different perspective from constructing the graphs as opposed to accepting the results of your graphing calculator.
The calculator is a valuable tool for checking yourself, and in some cases where a function cannot be reasonably constructed using standard techniques it becomes necessary. But that is not the case here.
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Self-critique (if necessary): good explanation.
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Self-critique Rating: 3
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Question: `q006. Make a table of y vs. x for y = `sqrt(x). [note: `sqrt(x) means 'the square root of x']. Graph y = `sqrt(x) between x = 0 and x = 3.
Would you say that the graph is increasing or decreasing?
Does the steepness of the graph change and if so, how?
Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?
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Your solution:
I used a calculator for this, and the corresponding x values 0, 1, 2, 3, for the graph points are (0,0), (1,1), (2,1.414), (3. 1.732), (4,2). From these points you can see the y values decreasing, hence the graph is decreasing rate.
The graph would be increasing at a decreasing rate.
If the graph represents the profile of a hill, the hill starts out very steep but gets easier and easier to climb. You are still climbing but you go up by less with each step, so the rate of increase is decreasing.
confidence rating #$&*: 2 (little hard)
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Self-critique (if necessary): good explanation.
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Self-critique Rating: 3
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Question: `q007. Make a table of y vs. x for y = 5 * 2^(-x). Graph y = 5 * 2^(-x) between x = 0 and x = 3.
Would you say that the graph is increasing or decreasing?
Does the steepness of the graph change and if so, how?
Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?
confidence rating #$&*: 2
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Your solution:
using a^(-b) = 1 / (a^b).
2^-2 = 1 / (2^2) = 1/4, so 5 * 2^-2 = 5 *1/4 = 5/4.
5* 2^-3 = 5 * (1 / 2^3) = 5 * 1/8 = 5/8.
Hence since the digits are decreasing for the y value, so can assume that the graph is at a decreasing rate.
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Self-critique (if necessary): good Q and A.
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Self-critique Rating: 3
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Question: `q008. Suppose you stand still in front of a driveway. A car starts out next to you and moves away from you, traveling faster and faster.
If y represents the distance from you to the car and t represents the time in seconds since the car started out, would a graph of y vs. t be increasing or decreasing?
Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?
confidence rating #$&*: 2
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Your solution:
Because the speed of the car increases so it goes further each second, it would proportionally effect the distance thus go in a positive slope and overall hence the graph would be increasing at an increasing rate.
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Self-critique (if necessary): good Q and A.
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Self-critique Rating: 3
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Question: `q009. As you saw above, on the interval from x = -3 to x = 3 the graph of y = x^2 is decreasing at a decreasing rate up to x = 0 and increasing at an increasing rate beyond x = 0.
How would you describe the behavior of the graph of y = (x - 1)^2 between x = -3 and x = 3?
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Your solution:
The behavior of the graph once graphs shows a negative correlation to y axis.
confidence rating #$&*:2
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Self-critique Rating: 2
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See my note about using the calculator to construct these graphs.
I'm going to ask you to resubmit this document, for reasons related to the graphing techniques you used, and also because you deleted parts of the original document (specifically the given solutions). The basic idea is never to delete anything from the document as it is presented, unless you are instructed to do so. There are a number of reasons for this, but the primary one is that your posted document should include the given solutions for your later reference.
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