#$&* course MTH 163 9/22/12 -- 2:15 PM
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Given Solution: Shifting the point -1 units in the horizontal direction we end up at the point (-3 + (-1), -1) = (-4, -1). Shifting the point 3 units in the vertical direction we end up at the point (-3, -1 + ( 3)) = (-3, 2). The point (-3, -1) is -1 units from the x axis. If the point is moves 4 times further from the x axis, the y coordinate will become 4 * -1 = -4. The x coordinate will not change. So the coordinates of the new point will be (-3, -4). If you then shift the resulting point -1 units in the horizontal direction, it will end up at (-3 + (-1), -4) = (-4, -4). If you shift this new point 3 units in the vertical direction, it will end up at (-4, -4 + 3) = (-4, -1). NOTE: We can express this sequence of transformations in a single step as (-3 + (-1), 4 * -1 + 3) = (-4, -1). Self-critique: Graphing was mathematically precise, just need to focus on applying proper signs when listing the points. ------------------------------------------------ Self-critique rating: Question `q002: Starting with the point P = (0, 0): Sketch the point you get if you shift this point -1 units in the horizontal direction. What are the coordinates of your point? Sketch the point you get if you shift the original point 3 units in the vertical direction. What are the coordinates of your point? Sketch the point you get if you move the original point 4 times as far from the x axis. What are the coordinates of your point? If you move the original point 4 times as far from the x axis, then shift the resulting point -1 units in the horizontal direction, and finally shift the point 3 units in the vertical direction, what are the coordinates of the final point? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your Solution: - Sketch the point you get if you shift this point -1 units in the horizontal direction. What are the coordinates of your point? -> (-1, 0) - - Sketch the point you get if you shift the original point 3 units in the vertical direction. What are the coordinates of your point? -> (0, 3) - - Sketch the point you get if you move the original point 4 times as far from the x axis. What are the coordinates of your point? -> (0, 4) - If you move the original point 4 times as far from the x axis, then shift the resulting point -1 units in the horizontal direction, and finally shift the point 3 units in the vertical direction, what are the coordinates of the final point? -> (-1, 7) - confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Shifting the point -1 units in the horizontal direction we end up at the point (0 + (-1), 0) = (-1, 0). Shifting the point 3 units in the vertical direction we end up at the point (0, 0 + ( 3)) = (0, 3). The point (0, 0) is 0 units from the x axis. If the point is moves 4 times further from the x axis, the y coordinate will be 4 * 0 = 0. The x coordinate will not change. So the coordinates of the new point will be (0, 0). If you then shift the resulting point -1 units in the horizontal direction, it will end up at (-1, 0). If you shift this new point 3 units in the vertical direction, it will end up at (-1, 3) Self-critique: Small mistake was made with the third and fourth points, but the mistake is understood now. Noted that 4 *0 = 0, but one had performed the action as if it started one unit about the origin to get (0,4) and (-1, 7). Duly noted that the points would instead become (-1, 0) and finally (-1,3). ------------------------------------------------ Self-critique rating: Question `q003: Plot the points (0, 0), (-1, 1) and (1, 1) on a set of coordinate axes. Now plot the points you get if you move each of these points 4 times further from the x axis, and put a small circle around each point. What are the coordinates of your points? Plot the points that result if you shift each of your three circled points -1 units in the x direction. Put a small 'x' through each point. What are the coordinates of your points? Plot the points that result if you shift each of your three new points (the ones with the x's) 3 units in the y direction. Put a small '+' through each point. What are the coordinates of your points? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your Solution: - Now plot the points you get if you move each of these points 4 times further from the x axis, and put a small circle around each point. What are the coordinates of your points? - (-1, 4) , (1, 4) , (0, 0) - Plot the points that result if you shift each of your three circled points -1 units in the x direction. Put a small 'x' through each point. What are the coordinates of your points? - (-2, 4) , (0, 4) , (-1, 0) - Plot the points that result if you shift each of your three new points (the ones with the x's) 3 units in the y direction. Put a small '+' through each point. What are the coordinates of your points? - (-2, 7) , (-1, 3) , (0, 7) confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Moving each point 4 times further from the x axis: The point (0, 0) is 0 units from the x axis. Multiplying this distance by 4 still gives you 0. So the point (0, 0) will remain where it is. The points (-1, 1) and (1, 1) are both 1 unit above the x axis. Multiplying this distance by 4 gives us 4 * 1 = 4. The x coordinates will not change, so our new points are (-1, 4) and (1, 4). At this stage our three points are (-1, 4) (0, 0) (1, 4) Horizontally shifting each point -1 units, our x coordinates all change by -1. We therefore obtain the points (-1 + (-1), 4) = (-2, 4), (0 + -1, 0) = (-1, 0) and ((1 + (-1), 4) = ( 0, 4), so our points are now (-2, 4) (-1, 0) ( 0, 4) Vertically shifting each point 3 units, our y coordinates all change by 3. We therefore obtain the points (-2, 4 + 3) = (-2, 7) (-1, 0 + 3) = (-1, 3) and ( 0, 4 + 3) = ( 0, 7) Self-critique: Had a bit of difficulty consulting the directions, mainly for the (x times as far) comments, but one believes it shouldn’t hinder graphing. ------------------------------------------------ Self-critique rating: Question `q004: On the coordinate axes you used in the preceding, sketch the parabola corresponding to the three basic points (0, 0), (-1, 1) and (1, 1). Then sketch the parabola corresponding to your three circled basic points. Then sketch the parabola corresponding to three basic points you indicated with 'x's'. Finally sketch the parabola corresponding to the three basic points you indicated with '+'s'. Describe how each parabola is related to the one before it. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your Solution: confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Your 'circled-points' parabola will be narrower than the original parabola through (-1, 1), (0, 0) and (1, 1). In fact, each point on the 'circled-points' parabola will lie 4 times further from the x axis than the point on the original parabola. Your 'x'-points parabola will have the same shape as your 'circled-points' parabola, but will lie to the right or left of that parabola, having been shifted -1 units in the horizontal direction. Your '+'-points parabola will have the same shape as the 'x-points' parabola (and the 'circled-point' parabola), but will lie above or below that parabola, having been shifted 3 units in the vertical direction. Self-critique: ------------------------------------------------ Self-critique rating: ********************************************* Question: `q005. Begin to solve the following system of simultaneous linear equations by first eliminating the variable which is easiest to eliminate. Eliminate the variable from the first and second equations, then from the first and third equations to obtain two equations in the remaining two variables: 2a + 3b + c = 128 60a + 5b + c = 90 200a + 10 b + c = 0. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: - 58a + 2b = -38 - 198a + 7b = 128 confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: The variable c is most easily eliminated. We accomplish this if we subtract the first equation from the second, and the first equation from the third, replacing the second and third equations with respective results. Subtracting the first equation from the second, are left-hand side will be the difference of the left-hand sides, which is 2d eqn - 1st eqn left-hand side: (60a + 5b + c )- (2a + 3b + c ) = 58 a + 2 b. The right-hand side will be the difference 90 - 128 = -38, so the second equation will become new' 2d equation: 58 a + 2 b = -38. The 'new' third equation by a similar calculation will be 'new' third equation: 198 a + 7 b = -128. You might well have obtained this system, or one equivalent to it, using a slightly different sequence of calculations. (As one example you might have subtracted the second from the first, and the third from the second). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): In the second “new” equation, how is the difference on the right side of the equals sign -128? I thought that was just 128 - 0 = +128? ------------------------------------------------ Self-critique rating:
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Given Solution: Neither variable is as easy to eliminate as in the last problem, but the coefficients of b are significantly smaller than those of a. So here we choose eliminate b. It would also have been OK to choose to eliminate a. To eliminate b we will multiply the first equation by -7 and the second by 2, which will make the coefficients of b equal and opposite. The first step is to indicate the multiplications: -7 * ( 58 a + 2 b) = -7 * -38 2 * ( 198 a + 7 b ) = 2 * (-128) Doing the arithmetic we obtain -406 a - 14 b = 266 396 a + 14 b = -256. Adding the two equations we obtain -10 a = 10, so we have a = -1. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q007. Having obtained a = -1, use either of the equations 58 a + 2 b = -38 198 a + 7 b = -128 to determine the value of b. Check that a = -1 and the value obtained for b are validated by the other equation. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: - Easy algebra. Simply substitute every variable where a = -1. 58(-1) + 2 b = -38 - -58 + 2b = -38 - 2b = 20 - b = 10 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: You might have completed this step in your solution to the preceding problem. Substituting a = -1 into the first equation we have 58 * -1 + 2 b = -38, so 2 b = 20 and b = 10. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q008. Having obtained a = -1 and b = 10, determine the value of c by substituting these values for a and b into any of the 3 equations in the original system 2a + 3b + c = 128 60a + 5b + c = 90 200a + 10 b + c = 0. Verify your result by substituting a = -1, b = 10 and the value you obtained for c into another of the original equations. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: - I’ll pick the first equation to solve for c. All of the c variables are 1, so it shouldn’t be too troubling. First I’ll plug in the values for ‘a’ and ‘b’ (-1, 10 respectively) and isolate the remaining variable, ‘c’, then solve accordingly. - 2(-1) + 3(10) + c = 128 - -2 + 30 + c = 128 - 28 + c = 128 - C = 100 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Using first equation 2a + 3b + c = 128 we obtain 2 * -1 + 3 * 10 + c = 128, which we easily solve to get c = 100. Substituting these values into the second equation, in order to check our solution, we obtain 60 * -1 + 5 * 10 + 100 = 90, or -60 + 50 + 100 = 90, or 90 = 90. We could also substitute the values into the third equation, and will again obtain an identity. This would completely validate our solution. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): When I first saw the problem for number 5, I completely solved it from memory without stopping at the points required for 5 and 6. However, I ended up with the following solutions: a = 1 b = 10 c = 96 The first equation checked perfectly, but the others were obviously a different story. ------------------------------------------------ Self-critique rating: ********************************************* Question: `q009. The graph you sketched in a previous assignment contained the given points (1, -2), (3, 5) and (7, 8). We are going to use simultaneous equations to obtain the equation of that parabola. A graph has a parabolic shape if its the equation of the graph is quadratic. The equation of a graph is quadratic if it has the form y = a x^2 + b x + c. y = a x^2 + b x + c is said to be a quadratic function of x. To find the precise quadratic function that fits our points, we need only determine the values of a, b and c. As we will discover, if we know the coordinates of three points on the graph of a quadratic function, we can use simultaneous equations to find the values of a, b and c. The first step is to obtain an equation using the first known point. What equation do we get if we substitute the x and y values corresponding to the point (1, -2) into the form y = a x^2 + b x + c? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: -2 = a(1)^2 + b(1) + c confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: We substitute y = -2 and x = 1 to obtain the equation -2 = a * 1^2 + b * 1 + c, or a + b + c = -2. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q010. If a graph of y vs. x contains the points (1, -2), (3, 5) and (7, 8), as in the preceding question, then what two equations do we get if we substitute the x and y values corresponding to the point (3, 5), then the point (7, 8) into the form y = a x^2 + b x + c? (each point will give us one equation) YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: - (1, -2) becomes a + b + c = -2 - (3, 5) becomes 9a + 3b + c = 5 - (7, 8) becomes 49a + 7b + c = 8 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Using the second point we substitute y = 5 and x = 3 to obtain the equation 5 = a * 3^2 + b * 3 + c, or 9 a + 3 b + c = 5. Using the third point we substitute y = 8 and x = 7 to obtain the equation 8 = a * 7^2 + b * 7 + c, or 49 a + 7 b + c = 8. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q011. If a graph of y vs. x contains the points (1, -2), (3, 5) and (7, 8), as was the case in the preceding question, then we obtain three equations with unknowns a, b and c. You have already done this. Write down the system of equations we got when we substituted the x and y values corresponding to the point (1, -2), (3, 5), and (7, 8), in turn, into the form y = a x^2 + b x + c. Solve the system to find the values of a, b and c. What is the solution of this system? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: My approximations were as follows: a = ( - . 4583 ) b = 5.3332 c =( - 6.8749 ) After trial and error, I see that it’s easier to, when combining systems of equations, have the greater base of ‘a’ on top. So theoretically it’s (2) and (1) in this case, rather than (1) and (2). - I combined the first two equations to get 8a + 2b = 7 for my equation (4) and then the first and last to get 48a + 6b = 10 for my equation (5). To get the first variable solved, I combined the two (and evaluated that the first equation must be multiplied by 6 and the second by 2 in order to eliminate ‘b’) to get a = - . 4583. To solve for the second variable, ‘b’, I substituted the variable ‘a’ in equation (4). 8( - . 4583) + 2b = 7 then isolated/simplified the variable to get b = 5.3332. To solve for the final variable, ‘c’, I substituted the variables ‘a’ and ‘b’ into equation (1). (- . 4583) + (5.3332) + c = -2 (4.8749) + c = -2 c = -6.8749. The variables were checked twice for accuracy. The only problem was that, when substituting the variables and their values in equation (3) the result is 8.0008. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: The system consists of the three equations obtained in the last problem: a + b + c = -2 9 a + 3 b + c = 5 49 a + 7 b + c = 8. This system is solved in the same manner as in the preceding exercise. However in this case the solutions don't come out to be whole numbers. The solution of this system, in decimal form, is approximately a = - 0.45833, b = 5.33333 and c = - 6.875. If you obtained a different solution, you should show your solution. Start by indicating the system of two equations you obtained when you eliminated c, then indicate what multiple of each equation you put together to eliminate either a or b. ADDITIONAL DETAILS ON SOLUTION OF SYSTEM You should have enough practice by now to be able to solve the system; however signs can trip us all up, and I've decided to append the following: The second equation minus the first gives us 8a + 2 b = 7. To avoid a common error in subtracting these questions, note that the right-hand sides of these equations are 5 and -2, and that 5 - (-2) = 5 + 2 = 7. It is very common for students (and the rest of us as well) to get a little careless and calculate the right-hand side as 5 - 2 = 3. The third equation minus the first gives us 48 a + 6 b = 10 (again the right-hand side can trip us up; 8 - (-2) = 10. I often see the incorrect calculation 8 - 2 = 6). Now we solve these two equations, 8 a + 2 b = 7 and 48 a + 6 b = 10: If you subtract 3 times the first from the second you will get 24 a = -11, so that a = -.45833. Substituting this into 8 a + 2 b = 7 and solving for b you get b = 5.33333. Substituting these values of a and b into any of the three original equations you get c = -6.875. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Are my decimals accurate enough to be considered right? ------------------------------------------------ Self-critique rating:
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Given Solution: Substituting the values of a, b and c into the given form we obtain the equation y = - 0.45833 x^2 + 5.33333 x - 6.875. When we substitute 1 into the equation we obtain y = -.45833 * 1^2 + 5.33333 * 1 - 6.875 = -2. When we substitute 3 into the equation we obtain y = -.45833 * 3^2 + 5.33333 * 3 - 6.875 = 5. When we substitute 5 into the equation we obtain y = -.45833 * 5^2 + 5.33333 * 5 - 6.875 = 8.33333. When we substitute 7 into the equation we obtain y = -.45833 * 7^2 + 5.33333 * 7 - 6.875 = 8. Thus the y values we obtain for our x values yield the points (1, -2), (3, 5) and (7, 8). These are the points we used to obtain the formula. We also get the additional point (5, 8.33333). NOTE THAT ADDITIONAL QUESTIONS RELATED TO THIS EXERCISE CONTINUE IN q_a_ ASSIGNMENT FOR ASSIGNMENT 3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: "" " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: "" " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!