course Phy 231 end programĸza
......!!!!!!!!...................................
19:14:29 `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.
......!!!!!!!!...................................
RESPONSE --> First pick a range for your x values such as -3,-2,-1,0,1,2,3. Place these values in a column labeled 'x'. Compute the 'y' values by substituting x values into the equation. For x = -3 you get y = 3(-3) -4 which is y= -13 Put this value in another column labeled y, but on the same line with the corresponding x value which in this case is -3. For x = -2, you get y= -10. Again place these values in the appropriate columns in the right place. For the rest of the y values you get: -7, -4, -1,2,5 Place these in their correct places in the y-column. x | y ------------ -3 | -13 -2 | -10 -1 | -7 0 | -4 1 | -1 2 | 2 3 | 5 This graph is a straight line. It crosses the Y-axis at point (0,-4). We get this by setting X equal to zero, but if you notice x = 0 in our chart so the corresponding y-value is where it crosses the y-axis. To find where it crosses the x-axis set y = 0 and solve the equation to get (4/3). So the line crosses the x-axis at point ( 4/3, 0 ). confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:14:52 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.
......!!!!!!!!...................................
RESPONSE --> OK. self critique assessment: 2
.................................................
......!!!!!!!!...................................
19:15:31 `q002. Does the steepness of the graph in the preceding exercise (of the function y = 3x - 4) change? If so describe how it changes.
......!!!!!!!!...................................
RESPONSE --> No, lines have a uniform slope which means they have a uniform steepness. This can also be confirmed by looking at the graph. confidence assessment: 2
.................................................
......!!!!!!!!...................................
19:15:37 The graph forms a straight line with no change in steepness.
......!!!!!!!!...................................
RESPONSE --> self critique assessment: 2
.................................................
......!!!!!!!!...................................
19:18:27 `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)?
......!!!!!!!!...................................
RESPONSE --> To get the slope you pick two points and use the slope equation which is (y2 - y1) / (x2 - x1) picking any two combinations of x and y values. (-10 minus -13) / (-2 minus -3 ) = 3/1 The slope is 3/1 or 3. confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:18:46 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.
......!!!!!!!!...................................
RESPONSE --> OK self critique assessment: 2
.................................................
......!!!!!!!!...................................
19:22:35 `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?
......!!!!!!!!...................................
RESPONSE --> x | y --------- 0 | 0 1 | 1 2 | 4 3 | 9 Yes it is increasing with each point. Yes the steepness changes, the slope will be different depending on what two points you choose. As x increases the steepness of the graph increases. The graph is increasing at an increasing rate because the slope is continually increasing. confidence assessment: 2
.................................................
......!!!!!!!!...................................
19:22:54 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 STUDENT QUESTION: I understand increasing...im just not sure at what rate...how do you determine increasing at an increasing rate or a constant rate? INSTRUCTOR RESPONSE: Does the y value increase by the same amount, by a greater amount or by a lesser amount every time x increases by 1? In this case the increases get greater and greater. So the graph increases, and at an increasing rate. *&*&.
......!!!!!!!!...................................
RESPONSE --> OK self critique assessment: 2
.................................................
......!!!!!!!!...................................
19:24:51 `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?
......!!!!!!!!...................................
RESPONSE --> x | y --------- -3 | 9 -2 | 4 -1 | 1 0 | 0 This graph is decreasing, the values are getting smaller as you move left to right. Yes, the steepness declines because the slope is decreasing at a decreasing rate. The graph is decreasing at a decreasing rate. confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:25:04 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.
......!!!!!!!!...................................
RESPONSE --> OK self critique assessment: 2
.................................................
......!!!!!!!!...................................
19:27:10 `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?
......!!!!!!!!...................................
RESPONSE --> x | y --------- 0 | 0 1 | 1 2 | 1.414 3 | 1.732 The graph is increasing because the values from left to right are increasing. The steepness changes, it is getting less steep as the graph increases. The graph is increasing at an increasing rate. confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:27:29 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.
......!!!!!!!!...................................
RESPONSE --> OK self critique assessment: 2
.................................................
......!!!!!!!!...................................
19:31:20 `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?
......!!!!!!!!...................................
RESPONSE --> x | y --------- 0 | 5 1 | 1/2 2 | 1/4 3 | 1/8 The graph is decreasing at a decreasnig rate. Yes, the steepness is decreasing as the graph decreases. confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:32:31 ** 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. **
......!!!!!!!!...................................
RESPONSE --> I did my algebra wrong and got the wrong points, but genreally the right description of the graph. self critique assessment: 1
.................................................
......!!!!!!!!...................................
19:33:11 `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?
......!!!!!!!!...................................
RESPONSE --> The graph would be increasing at an increasing rate. The distance is increasing faster and faster as the car gains speed. confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:33:22 ** 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. **
......!!!!!!!!...................................
RESPONSE --> OK self critique assessment: 2
................................................."