#$&* course Mth 163 12/17 10 022. `query 22
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Given Solution: `a** x^-p = 1 / x^p. As x gets closer to 0, x^p gets closer to 0. Dividing 1 by a number which gets closer and closer to 0 gives us a result with larger and larger magnitude. There is no limit to how close x can get to 0, so there is no limit to how many times x^p can divide into 1. This results in y = x^p values that approach infinite distance from the x axis. The graph therefore approaches a vertical limit. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating: ok ********************************************* Question: `qExplain why the function y = (x-h)^-p has a vertical asymptote at x = h. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: When you solve this, you would get y = 1 / (x-)^p. Doing this it gets closer to zero so x = h and shifts vertically closer to zero. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** (x-h)^-p = 1 / (x-h)^p. As x gets closer to h, (x-h)^p gets closer to 0. Dividing 1 by a number which gets closer and closer to 0 gives us a result with larger and larger magnitude. There is no limit to how close x can get to h, so there is no limit to how many times (x-h)^p can divide into 1. This results in y = (x-h)^p values that approach infinite distance from the x axis as x approaches h. The graph therefore approaches a vertical limit. This can also be seen as a horizontal shift of the y = x^-p function. Replacing x by x - h shifts the graph h units in the x direction, so the asymptote at x = 0 shifts to x = h. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Can also be a horizontal shift, shifts graph h units in the x direction. ------------------------------------------------ Self-critique rating: ok ********************************************* Question: `qExplain why the function y = (x-h)^-p is identical to that of x^-p except for the shift of h units in the x direction. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: It would be the same except for the shift in the graph for the h value. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aSTUDENT ANSWER: You end up with the exact same y values but at the different position of x changed by the h value. INSTRUCTOR COMMENT: Good start. More specifically the x value at which a given y value occurs is shifted h units, so that for example y = x^p is zero when x = 0, but y = (x - h)^p is zero when x = h. To put this as a series of questions (you are welcome to insert answers to these questions, using #$&* before and after each insertion):: Assume that p is positive. For what value of x is x^p equal to zero? For what value of x is (x - 5)^p equal to zero? For what value of x is (x - 1)^p equal to zero? For what value of x is (x - 12)^p equal to zero? For what value of x is (x - h)^p equal to zero? For example, the figure below depicts the p = 3 power functions x^3, (x-1)^3 and (x-5)^3. Assume now that p is negative. For what value of x does the graph of y = x^p have a vertical asymptote? For what value of x does the graph of y = (x-1)^p have a vertical asymptote? For what value of x does the graph of y = (x-5)^p have a vertical asymptote? For what value of x does the graph of y = (x-12)^p have a vertical asymptote? For what value of x does the graph of y = (x-h)^p have a vertical asymptote? For example, the figure below depicts the p = -3 power functions x^-3 and (x-5)^-3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating: ok ********************************************* Question: `qGive your table (increment .4) showing how the y = x^-3 function can be transformed first into y = (x - .4) ^ -3, then into y = -2 (x - .4) ^ -3, and finally into y = -2 (x - .4) ^ -3 + .6. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: X y = x^3 y = (x-.4)^-3 y = -2(x-.4)^-3 y = -2(x-.4)^-3+.6 -.4 -.064 -1.953 3.906 4.50625 0 0 -15.625 31.25 31.85 .4 .064 infinity infinity - infinity .8 .512 15.625 -31.25 -30.65 .12 .001728 -45.5539 91.1078 91.7079 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** The table is as follows (note that column headings might not line up correctly with the columns): x y = x^3 y = (x-0.4)^(-3) y = -2 (x-0.4)^(-3) y = -2 (x-0.4)^(-3) + .6 -0.8 -1.953 -0.579 1.16 1.76 -0.4 -15.625 -1.953 3.90 4.50 0 div by 0 -15.625 31.25 32.85 0.4 15.625 div by 0 div by 0 div by 0 0.8 1.953 15.625 -31.25 -30.65 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating: ok ********************************************* Question: `qExplain how your table demonstrates this transformation and describe the graph that depicts the transformation. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: y = (x-.4)^-3 shifts the graph .4 to the right. y = -2(x-.4)^-3 shifts the graph .4 to the right and stretches vertical by -2 y = -2(x-.4)^-3 + .6 shifts the graph .6 vertically. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a y = x^-3 transforms into y = (x - .4)^-3, shifting the basic points .4 unit to the right. The vertical asymptote at the y axis (x = 0) shifts to the vertical line x = .4. The x axis is a horizontal asymptote. y = -2 (x - .4)^-3 vertically stretches the graph by factor -2, moving every point twice as far from the x axis and also to the opposite side of the x axis. This leaves the vertical line x = .4 as a vertical asymptote. The x axis remains a horizontal asymptote. y = -2 ( x - .4)^-3 + 6 vertically shifts the graph +6 units. This has the effect of maintaining the shape of the graph but raising the horizontal asymptote to x = 6. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating: ok ********************************************* Question: `qDescribe your graphs of y = x ^ .5 and y = 3 x^.5. Describe how your graph depicts the ratios of y values between the two functions. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The graph y = 3 x^.5 compared to y = x^.5 stretches vertically by a factor of 3. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a*&*& This is a power function y = x^p with p = .5. The basic points of y = x^.5 are (0, 0), (.5, .707), (1, 1), (2, 1.414). • Attempting to find a basic point at x = -1 we find that -1^-.5 is not a real number, leading us to the conclusion that y = x^.5 is not defined for negative values of x. • The graph therefore begins at the origin and increases at a decreasing rate. • However since we can make x^.5 as large as we wish by making x sufficiently large, there is no horizontal asymptote. y = 3 x^.5 vertically stretches the graph of y = x^.5 by factor 3, giving us basic points (0, 0), (.5, 2.12), (1, 3) and (2, 4.242). • This graph is also increasing at a decreasing rate, staying 3 times as far from the x axis as the graph of the original y = x^.5. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating: ok ********************************************* Question: `qExplain why the graph of A f(x-h) + k is different than the graph of A [ f(x-h) + k ], and describe the difference. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The difference would be because the graph of A [ f(x-h) + h] you would have to do what is in the brackets first, which would give you a different shift and stretch than the first graph. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** The first graph is obtained from y = f(x) by first vertically stretching by factor A, then horizontall shifting h units and finally vertically shifting k units. The graph of A [f(x-h) + k] is obtained by first doing what is in brackets, horizontally shifting h units then vertically shifting k units before doing the vertical stretch by factor A. Thus the vertical stretch applies to the vertical shift in addition to the values of the function. This results in different y coordinates and a typically a very different graph. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating: ok If you understand the assignment and were able to solve the previously given problems from your worksheets, you should be able to complete most of the following problems quickly and easily. If you experience difficulty with some of these problems, you will be given notes and we will work to resolve difficulties. ********************************************* Question: Explain why the function y = (x - 3) ^ (-2) has a vertical asymptote at x = +3, while the function y = (x + 3)^(-2) has a vertical asymptote at x = -3. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The first would have a vertical asymptote of x = 3 because in the parentheses you have (x-3) which the zero would be 3. For the second function, it would be -3 because in the parentheses there is (x + 3) so the zero would be -3. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ********************************************* Question: Use a table containing appropriate values to demonstrate the transformation of the y = x^(.5) function into the y = 2 (x - 3) ^ .5 - 1 function, as a series of transformations in which the original function is transformed first to y = 2 x^.5, then to y = 2 ( x - 3) ^ .5, then finally to the final form. Briefly describe the original graph, and how the graph changes with each successive transformation. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: X y= x^.5 y = 2(x-3) ^.5-1 y = 2x^.5 y = 2(x-3)^.5 -.4 -.632 error -1.26 error 0 0 error 0 error .4 .632 1 1.26 error confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ********************************************* Question: Starting with the function y = f(x), describe the series of graphical transformations required to transform its graph to the graph of y = A f(x - h) + k. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: This graph would have a stretch of A with a shift of h, then shift of k. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ------------------------------------------------ Self-critique rating: Query Add comments on any surprises or insights you experienced as a result of this assignment. " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: Query Add comments on any surprises or insights you experienced as a result of this assignment. " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!