course Phy 121 ۬JzIشĖHN{Student Name:
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21:44:17 `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 email your work to the instructor. 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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RESPONSE --> The first column in the table is labeled x and the second column in the table is labeled y, as follows: y=3x-4 x y -3 -13 -2 -10 -1 -7 0 -4 1 -1 2 2 3 5 In order to find the y values, you must substitute the x into the equation. Then graph these coordinates. The points where the graph goes through the x and y axes are: intercepts y-axis at (0,-4) and the x-axis at about (1,0)
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21:47:43 The graph goes through the x axis when y = 0 and through the y axis when x = 0. The x-intercept is therefore when 0 = 3x - 4, so 4 = 3x and x = 4/3. The y-intercept is when y = 3 * 0 - 4 = -4. Thus the x intercept is at (4/3, 0) and the y intercept is at (0, -4). Your graph should confirm this.
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RESPONSE --> Only when y=0 does the graph go through the x-axis, and only when x=0 does the graph go through the y-axis. The graph goes through the x-axis when: 0=3x-4 The graph goes through the y-axis when: y=3(0) - 4 When graphing, this should be apparent.
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21:49:12 `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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RESPONSE --> No because the slope doesn't change.
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21:49:46 The graph forms a straight line with no change in steepness.
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RESPONSE --> Ok, I understand this. The line just continues to grow longer.
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21:51:05 `q003. What is the slope of the graph of the preceding two exercises (the function ia y = 3x - 4;slope is rise / run between two points of the graph)?
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RESPONSE --> The slope, being rise/run, is 3/1. The slope of this graph must be positive because the line lies to the right side.
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21:53:30 Between any two points of the graph rise / run = 3. For example, when x = 2 we have y = 3 * 2 - 4 = 2 and when x = 8 we have y = 3 * 8 - 4 = 20. Between these points the rise is 20 - 2 = 18 and the run is 8 - 2 = 6 so the slope is rise / run = 18 / 6 = 3. Note that 3 is the coefficient of x in y = 3x - 4. Note the following for reference in subsequent problems: The graph of this function is a straight line. The graph increases as we move from left to right. We therefore say that the graph is increasing, and that it is increasing at constant rate because the steepness of a straight line doesn't change.
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RESPONSE --> No matter how far you go with the coordinates, the slope will always stay the same. In y=3x-4, 3 is the coefficient of x. Since the function is in a straight line, we can see that the graph increases when looking left to right. (therefore it is increasing at a constant rate because the slope doesn't change)
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21:59:22 `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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RESPONSE --> The table is: x y 0 0 1 1 2 4 3 9 Thus, when you graph, the graph is increasing.The steepness of the graph does not stay steady. It goes out from the first point to the second point, then up to the third point, and then up even more to the fourth point. I would say the graph is increasing at a decreasing rate.
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22:00:21 Graph points include (0,0), (1,1), (2,4) and (3,9). The y values are 0, 1, 4 and 9, which increase as we move from left to right. The increases between these points are 1, 3 and 5, so the graph not only increases, it increases at an increasing rate.
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RESPONSE --> Correction: The graph is increasing at an increasing rate. This is because it continually increases as you move left to right.
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22:05:57 `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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RESPONSE --> The table is as follows. x y -3 9 -2 4 -1 1 0 0 When you graph these coordinates, the graph proves to be decreasing. The steepness is not continually steady. The line through the point curves downward slightly. I would say that the graph is decreasing at an increasing rate.
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22:08:01 From left to right the graph is decreasing (points (-3,9), (-2,4), (-1,1), (0,0) show y values 9, 4, 1, 0 as we move from left to right ). The magnitudes of the changes in x from 9 to 4 to 1 to 0 decrease, so the steepness is decreasing. Thus the graph is decreasing, but more and more slowly. We therefore say that the graph is decreasing at a decreasing rate.
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RESPONSE --> I accidentally put decreasing at an increasing rate, when I meant decreasing at a decreasing rate. I understand the concept of the decreasing/increasing at different rates.
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22:12:06 `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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RESPONSE --> Table: x y 0 0 1 1 2 ~1.4 3 ~1.7 When graphed, the graph is increasing. The steepness of the graph is unsteady. The graph is increasing at a decreasing rate.
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22:13:11 If you use x values 0, 1, 2, 3, 4 you will obtain graph points (0,0), (1,1), (2,1.414), (3. 1.732), (4,2). The y value changes by less and less for every succeeding x value. Thus the steepness of the graph is decreasing. The graph would be increasing at a decreasing rate.{}{} If the graph respresents 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. {}{}If your graph doesn't look like this then you probably are not using a consistent scale for at least one of the axes. If your graph isn't as desribed take another look at your plot and make a note in your response indicating any difficulties.
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RESPONSE --> I understand the concept of all this. Even though the line is increasing, it is doing so each time at a slower and slower rate.
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22:20:09 `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?
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RESPONSE --> y= 5 * 2^(-x) x y 0 5 1 2.5 2 1.25 3 .625 When graphed, the graph is decreasing. The steepness is not constant or the same. I would say the graph is decreasing at a decreasing rate.
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22:21:23 ** From basic algebra recall that a^(-b) = 1 / (a^b). So, for example: 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. Etc. The decimal equivalents of the values for x = 0 to x = 3 will be 5, 2.5, 1.25, .625. These values decrease, but by less and less each time. The graph is therefore decreasing at a decreasing rate. **
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RESPONSE --> For this problem, algebra must be used. The graph is decreasing at a decreasing rate, because the values for y decrease by less and less each time.
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22:25:32 `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?
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RESPONSE --> The graph would be increasing at an increasing rate. The amount of time continues to increase, and so does the distance.
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22:26:27 ** The speed of the car increases so it goes further each second. On a graph of distance vs. clock time there would be a greater change in distance with each second, which would cause a greater slope with each subsequent second. The graph would therefore be increasing at an increasing rate. **
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RESPONSE --> Speed increases as it goes further away each second. Also, time continually increases.
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