course Mth 163 I sent this message through an e-mail and after I did it I thought maybe I should send this through here.
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qa prelim ????O???R???? assignment #002 002. Describing Graphs qa initial problems 01-11-2008
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22:59:11 `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. 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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RESPONSE --> The point crosses the x axes at (1,0) and the y axes at (0,- 4). confidence assessment: 2
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23:06:31 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 --> OK $$$$$I see where the number goes into a fraction. It happened from where the 3 was multiplied by the 0.$$$$$ self critique assessment: 2
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23:07:39 `q002. Does the steepness of the graph in the preceding exercise (of the function y = 3x - 4) change? If so describe how it changes.
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RESPONSE --> Yes. It goes over to the right a little and makes the rest of the graph steep. confidence assessment: 2
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23:07:53 The graph forms a straight line with no change in steepness.
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RESPONSE --> Yes $$$$$If the numbers were different the line would be steeper but where there was no change you can see the straight line.$$$$$ self critique assessment: 2
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23:08:16 `q003. What is the slope of the graph of the preceding two exercises (the function ia y = 3x - 4;slope is rise / run between two points of the graph)?
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RESPONSE --> Yes $$$$$The rise/run is (4,1). You can see it as you look at the graph from left to right.$$$$$ confidence assessment: 2
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23:09:55 Between any two points of the graph rise / run = 3. For example, when x = 2 we have y = 3 * 2 - 4 = 2 and when x = 8 we have y = 3 * 8 - 4 = 20. Between these points the rise is 20 - 2 = 18 and the run is 8 - 2 = 6 so the slope is rise / run = 18 / 6 = 3. Note that 3 is the coefficient of x in y = 3x - 4. Note the following for reference in subsequent problems: The graph of this function is a straight line. The graph increases as we move from left to right. We therefore say that the graph is increasing, and that it is increasing at constant rate because the steepness of a straight line doesn't change.
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RESPONSE --> Yes $$$$$I see how the slope was determended by subtracting 20- 2=18. I don't understand where the 8-2=6 comes in to get the slope of 3.$$$$$
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23:13:25 `q004. Make a table of y vs. x for y = x^2. Graph y = x^2 between x = 0 and x = 3. Would you say that the graph is increasing or decreasing? Does the steepness of the graph change and if so, how? Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?
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RESPONSE --> Increasing. No. Increasing at an increasing rate. confidence assessment: 2
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23:15:06 `q005. Make a table of y vs. x for y = x^2. Graph y = x^2 between x = -3 and x = 0. Would you say that the graph is increasing or decreasing? Does the steepness of the graph change and if so, how? Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?
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RESPONSE --> At an increasing it rate the numbers change the same throughout the problem. At a constant rate it stays the same as we move from left to right. confidence assessment: 2
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23:15:45 From left to right the graph is decreasing (points (-3,9), (- 2,4), (-1,1), (0,0) show y values 9, 4, 1, 0 as we move from left to right ). The magnitudes of the changes in x from 9 to 4 to 1 to 0 decrease, so the steepness is decreasing. Thus the graph is decreasing, but more and more slowly. We therefore say that the graph is decreasing at a decreasing rate.
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RESPONSE --> A rate of 3 $$$$$I see where the graph is decreasing in the y axes from 9,4,2,and 0. I don't understand what the magnitude has to do with it. I can see the graph decreasing as you look at it.$$$$$
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23:20:45 `q006. Make a table of y vs. x for y = `sqrt(x). [note: `sqrt (x) means 'the square root of x']. Graph y = `sqrt(x) between x = 0 and x = 3. Would you say that the graph is increasing or decreasing? Does the steepness of the graph change and if so, how? Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?
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RESPONSE --> Increasing. Yes. By 1.7. Increasing at an increasing rate. confidence assessment: 2
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23:22:02 If you use x values 0, 1, 2, 3, 4 you will obtain graph points (0,0), (1,1), (2,1.414), (3. 1.732), (4,2). The y value changes by less and less for every succeeding x value. Thus the steepness of the graph is decreasing. The graph would be increasing at a decreasing rate. If the graph respresents the profile of a hill, the hill starts out very steep but gets easier and easier to climb. You are still climbing but you go up by less with each step, so the rate of increase is decreasing. If your graph doesn't look like this then you probably are not using a consistent scale for at least one of the axes. If your graph isn't as desribed take another look at your plot and make a note in your response indicating any difficulties.
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RESPONSE --> OK $$$$$I see how the graph changes less and less as you look at it. As you look at the numbers and points and the line it is a visual aid to see how the graph changes.$$$$$ self critique assessment: 2 &# This also requires a self-critique. &#
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23:24:52 `q007. Make a table of y vs. x for y = 5 * 2^(-x). Graph y = 5 * 2^(-x) between x = 0 and x = 3. Would you say that the graph is increasing or decreasing? Does the steepness of the graph change and if so, how? Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?
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RESPONSE --> Decreasing. Yes. It slopes down. Decreasing at a constant rate. confidence assessment: 2
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23:27:37 ** From basic algebra recall that a^(-b) = 1 / (a^b). So, for example: 2^-2 = 1 / (2^2) = 1/4, so 5 * 2^-2 = 5 * 1/4 = 5/4. 5* 2^-3 = 5 * (1 / 2^3) = 5 * 1/8 = 5/8. Etc. The decimal equivalents of the values for x = 0 to x = 3 will be 5, 2.5, 1.25, .625. These values decrease, but by less and less each time. The graph is therefore decreasing at a decreasing rate. **
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RESPONSE --> OK $$$$$I was some what confused when I looked at this one. I wasn't sure what the deal was with all the fractions. I got the decimal part right on one of them which was .625. The rest I didn't get at all. I am not sure what I did. I understood the part that the graph is decreasing at a decreasing rate.$$$$$
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23:30:38 `q008. Suppose you stand still in front of a driveway. A car starts out next to you and moves away from you, traveling faster and faster. If y represents the distance from you to the car and t represents the time in seconds since the car started out, would a graph of y vs. t be increasing or decreasing? Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?
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RESPONSE --> Increasing. Increasing at an increasing rate. confidence assessment: 2
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23:31:17 ** The speed of the car increases so it goes further each second. On a graph of distance vs. clock time there would be a greater change in distance with each second, which would cause a greater slope with each subsequent second. The graph would therefore be increasing at an increasing rate. **
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RESPONSE --> OK $$$$$The faster the car goes the more seconds that goes by. The longer the car runs the more it is going. It is always increasing with the seconds and the speed that it is going.$$$$$ self critique assessment: 2
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Your answers demonstrate that you understand some of these concepts, but you haven't included enough justification in your answers to demonstrate what you do and do not understand, and you did not self-critique in the prescribed manner. &#Please respond with a copy of this document, including my comments. Insert your revisions and/or questions and/or self- critiques, as appropriate, and mark them with $$$$. &# "