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17:05:31 Note that you can't use a calculator graph to document your solutions to these problems. You have to use the analytical methods as in the given solutions. Documentation is required on tests, and while you may certainly use the calculator to see symmetry, intercepts etc., you have to support your solutions with the algebraic details of why the graph looks the way it does.
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RESPONSE --> Ok Don't depend on the calculator for your work.
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17:19:06 query 2.2.34 / 10 (was 2.2.6). Point symmetric to (-1, -1) wrt x axis, y axis, origin. What point is symmetric to the given point with respect to each: x axis, y axis, the origin?
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RESPONSE --> The point symetrical to (-1, -1) on the x-axis is the point ( -1, 1) The point symetrical on the y axis is (1,1). The orgin (1,1)
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17:19:38 ** There are three points: The point symmetric to (-1, -1) with respect to the x axis is (-1 , 1). The point symmetric to (-1, -1) with respect to the y axis is y axis (1, -1) The point symmetric to (-1, -1) with respect to the origin is ( 1,1). **
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RESPONSE --> ok
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20:33:22 **** query 2.2.43 / 19 (was 2.2.15). Parabola vertex origin opens to left. **** Give the intercepts of the graph and tell whether the graph is symmetric to the x axis, to the y axis and to the origin. Explain how you determined the answer to each question.
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RESPONSE --> The intercepts of the graph are (0,0) It is symetric to the x axis x=0 making y=0 and when y=0 then x=0
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20:35:53 ** The graph intercepts both axes at the same point, (0,0) The graph is symmetric to the x-axis, with every point above the x axis mirrored by its 'reflection' below the x axis. **
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RESPONSE --> I like this definition of symmetry ""every point above the x axis is mirrored by it reflection below the x axis.""
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20:41:38 **** query 2.2.48 / 24 (was 2.2.20). basic cubic poly arb vert stretch **** Give the intercepts of the graph and tell whether the graph is symmetric to the x axis, to the y axis and to the origin. Explain how you determined the answer to each question.
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RESPONSE --> The graph intercepts are (1/2, -1/2) The graph ia symetric to the orgin.
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20:54:03 The graph s strictly increasing except perhaps at the origin where it might level off for just an instant, in which case the only intercept is at the origin (0, 0). The graph is symmetric with respect to the origin, since for every x we have f(-x) = - f(x). For example, f(2) = 8 and f(-2) = -8. It looks like f(1) = 1 and f(-1) = -1. Whatever number you choose for x, f(-x) = - f(x).
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RESPONSE --> The intercept is at the orgin (0,0 )
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21:13:33 **** query 2.2.62 / 40 (was 2.2.36). 4x^2 + y^2 = 4 **** List the intercepts and explain how you made each test for symmetry, and the results of your tests.
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RESPONSE --> 4x^2 + y^2 =4 4x^2 + 0^2 =4 4x^2 = 4 x^2= 1 x= +-1 4(0) + y^2 =4 0+ y^2 =4 y^2 = 4 y=+ - 2 Intercepts (1, 2) (-1, -2) (1,-2) (-1, 2) Symmetry checkFor x axis let y=-y; 4 x^2 + (-y) ^2 =4: 4x^2 +y =4 Symmetry for y axis. Let x= -x. 4(-x) + y^2 = 4; 4x^2 =y =4; Orgin symmetry 4(-x)^2 + (-y) ^2 =4; 4x^2 + y^2 =4.
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21:15:16 ** Starting with 4x^2 +y^2 = 1 we find the x intercept by letting y = 0. We get 4x^2 + 0 = 1 so 4x^2 = 1 and x^2=1/4 . Therefore x=1/2 or -1/2 and the x intercepts are (1/2,0) and ( -1/2,0). Starting with 4x^2 +y^2 = 1 we find the y intercept by letting x = 0. We get 0 +y^2 = 1 so y^2 = 1 and y= 1 or -1, giving us y intercepts (0,1) and (0,-1). To test for symmetry about the y axis we substitute -x for x. If there's no change in the equation then the graph will be symmetric to about the y axis. Substituting we get 4 (-x)^2 + y^2 = 1. SInce (-x)^2 = x^2 the result is 4 x^2 + y^2 = 1. This is identical to the original equation so we do have symmetry about the y axis. To test for symmetry about the x axis we substitute -y for y. If there's no change in the equation then the graph will be symmetric to about the x axis. Substituting we get 4 (x)^2 + (-y)^2 = 1. SInce (-y)^2 = y^2 the result is 4 x^2 + y^2 = 1. This is identical to the original equation so we do have symmetry about the x axis. To test for symmetry about the origin we substitute -x for x and -y for y. If there's no change in the equation then the graph will be symmetric to about the origin. Substituting we get 4 (-x)^2 + (-y)^2 = 1. SInce (-x)^2 = x^2 and (-y)^2 - y^2 the result is 4 x^2 + y^2 = 1. This is identical to the original equation so we do have symmetry about the origin. **
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RESPONSE --> My answer was different because my problem was equal to 4 instead of 1.
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21:31:25 **** query 2.2.68 / 46 (was 2.2.42). y = (x^2-4)/(2x) **** List the intercepts and explain how you made each test for symmetry, and the results of your tests.
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RESPONSE --> y = x^2-4/2x; x intercepts 0 = x^2 -4/2x x=0 y intercepts y=0^2 -4/2(0) y=-4/0 = +- 4 (0, -4) (0 4) Test for symmetry x axis; -y = x^2 -4/2x different Test for symmetry y axis; y = (-4)^2 -4/ 2(-x) Test for orgin symmetry; -y = (-x)^2 -4/2 (-x)
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21:33:37 ** We do not have symmetry about the x or the y axis, but we do have symmetry about the origin: To test for symmetry about the y axis we substitute -x for x. If there's no change in the equation then the graph will be symmetric to about the y axis. }Substituting we get y = ( (-x)^2 - 4) / (2 * (-x) ). SInce (-x)^2 = x^2 the result is y = -(x^2 - 4) / (2 x). This is not identical to the original equation so we do not have symmetry about the y axis. To test for symmetry about the x axis we substitute -y for y. If there's no change in the equation then the graph will be symmetric to about the x axis. Substituting we get -y = (x^2-4)/(2x) , or y = -(x^2-4)/(2x). This is not identical to the original equation so we do not have symmetry about the x axis. To test for symmetry about the origin we substitute -x for x and -y for y. If there's no change in the equation then the graph will be symmetric to about the origin. Substituting we get -y = ((-x)^2-4)/(2(-x)) SInce (-x)^2 = x^2 the result is -y = -(x^2-4)/(2x), or multiplying both sides by -1, our result is y = (x^2-4)/(2x). This is identical to the original equation so we do have symmetry about the origin. **
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RESPONSE --> ok
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