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Solutions for Session 7 Homework
See solutions for Problems: H1 | H2 | H3 | H4 | H5 | H6
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Problem H1 | |
a. | Any three points on the vertical axis will do -- for instance, (0,-3), (0,1), and (0,13). All the points with x-coordinate 0 are on the y-axis. |
b. | Any three points with the y-coordinate of 1 will do -- for instance, (-7,1), (0,1), and (12,1). All the points on this horizontal line have y-coordinate 1. |
c. | Any three points with x-coordinate 3 will do -- for instance, (3,-4), (3,0), and (3,11). A point is on the line v if its x-coordinate is 3. If its x-coordinate is anything other than 3, the point is not on the line. |
d. | The coordinates are (-4,2). |
e. | For instance, (-3,-3), (-2,-2), (0,0), (4,4), (15,15). All of these points are on the line y = x. |
<< back to Problem H1
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Problem H2 | |
a. | There are infinitely many points between the two lines -- for instance, (-32,-1), (-17,1) (0,0), (33,3), (155,3.5), (1000,3.9). |
b. | There are infinitely many points which are not between the two lines -- for instance, (-32,7), (-17,-15), (0,5), (33,7), (155,4.5), (1000,7.9). |
c. | A point is between the two lines if its y-coordinate is greater than -2 and less than 4. |
<< back to Problem H2
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Problem H3 | |
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A |
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B |
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C |
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D |
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E |
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F |
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G |
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(x,y) |
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(x + 3,
y - 2) |
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(-x,y) |
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(2x,2y) |
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(x - 1,
y + 2) |
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(y,-x) |
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(-y,x) |
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(2,1) |
(5,-1) |
(-2,1) |
(4,2) |
(1,3) |
(1,-2) |
(-1,2) |
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(-4,0) |
(-1,-2) |
(4,0) |
(-8,0) |
(-5,2) |
(0,4) |
(0,-4) |
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(-5,4) |
(-2,2) |
(5,4) |
(-10,8) |
(-6,6) |
(4,5) |
(-4,-5) |
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a. | Triangle B is the translation of triangle A 3 units to the right and 2 units down:
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b. | Triangle C is obtained by reflecting triangle A about the vertical axis:
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c. | Triangle D is obtained by stretching triangle A in both x- and y-direction by a factor of 2:
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d. | Triangle E is obtained by shifting triangle A 1 unit to the left and 2 units up:
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e. | Triangle F is obtained by reflecting triangle A about the vertical axis and then about the line y = x. Alternatively, it can also be obtained as a -90° rotation of triangle A about the origin (0,0).
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f. | Triangle G is obtained by reflecting triangle A about the horizontal axis and then about the line y = x. Alternatively, it can also be obtained as a 90° rotation of triangle A about the origin (0,0).
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<< back to Problem H3
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Problem H4 | |
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For example, (-2,3) becomes (-2,-3). In general, reflecting (x,y) about the horizontal axis yields (x,-y). In other words, the x-coordinate is unchanged while the y-coordinate is the negative of the original y-coordinate.
<< back to Problem H4
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Problem H5 | |
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For example, (-2,3) becomes (2,3). In general, reflecting (x,y) about the vertical axis yields (-x,y). In other words, the y-coordinate is unchanged while the x-coordinate is the negative of the original x-coordinate.
<< back to Problem H5
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Problem H6 | |
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For example (-2,3) becomes (3,-2). In general, if a point (x,y) is reflected about the line y = x, its new coordinates are (y,x). In other words, what used to be the x-coordinate becomes the y-coordinate, and what used to be the y-coordinate becomes the x-coordinate.
<< back to Problem H6
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