1.2E: Exercises
- Page ID
- 120078
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Exercises
In Exercises 1 - 20, graph the given relation.
- \(\ \{(-3,9),(-2,4),(-1,1),(0,0),(1,1),(2,4),(3,9)\}\)
- \(\ \{(-2,0),(-1,1),(-1,-1),(0,2),(0,-2),(1,3),(1,-3)\}\)
- [MOM] \(\ \{(m, 2 m) \mid m=0, \pm 1, \pm 2\}\)
- \(\ \left\{\left(\frac{6}{k}, k\right) \mid k=\pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6\right\}\)
- \(\ \left\{\left(n, 4-n^{2}\right) \mid n=0, \pm 1, \pm 2\right\}\)
- \(\ \{(\sqrt{j}, j) \mid j=0,1,4,9\}\)
- [MOM] \(\ \{(x,-2) \mid x>-4\}\)
- [MOM] \(\ \{(x, 3) \mid x \leq 4\}\)
- \(\ \{(-1, y) \mid y>1\}\)
- \(\ \{(2, y) \mid y \leq 5\}\)
- [MOM] \(\ \{(-2, y) \mid-3<y \leq 4\}\)
- [MOM] \(\ \{(3, y) \mid-4 \leq y<3\}\)
- \(\ \{(x, 2) \mid-2 \leq x<3\}\)
- \(\ \{(x,-3) \mid-4<x \leq 4\}\)
- [MOM] \(\ \{(x, y) \mid x>-2\}\)
- [MOM] \(\ \{(x, y) \mid x \leq 3\}\)
- \(\ \{(x, y) \mid y<4\}\)
- [MOM] \(\ \{(x, y) \mid x \leq 3, y<2\}\)
- [MOM] \(\ \{(x, y) \mid x>0, y<4\}\)
- \(\ \left\{(x, y) \mid-\sqrt{2} \leq x \leq \frac{2}{3}, \pi<y \leq \frac{9}{2}\right\}\)
In Exercises 21 - 30, describe the given relation using either the roster or set-builder method.
- [MOM]
In Exercises 31 - 36, graph the given line.
- \(\ x = −2\)
- \(\ x = 3\)
- \(\ y = 3\)
- \(\ y = −2\)
- \(\ x = 0\)
- \(\ y = 0\)
Some relations are fairly easy to describe in words or with the roster method but are rather difficult, if not impossible, to graph. Discuss with your classmates how you might graph the relations given in Exercises 37 - 40. Please note that in the notation below we are using the ellipsis, . . . , to denote that the list does not end, but rather, continues to follow the established pattern indefinitely. For the relations in Exercises 37 and 38, give two examples of points which belong to the relation and two points which do not belong to the relation.
- \(\ \{(x, y) \mid x \text { is an odd integer, and } y \text { is an even integer. }\}\)
- \(\ \{(x, 1) \mid x \text { is an irrational number }\}\)
- \(\ \{(1,0),(2,1),(4,2),(8,3),(16,4),(32,5), \ldots\}\)
- \(\ \{\ldots,(-3,9),(-2,4),(-1,1),(0,0),(1,1),(2,4),(3,9), \ldots\}\)
For each equation given in Exercises 41 - 52:
- Find the x- and y-intercept(s) of the graph, if any exist.
- Create a table of sample points on the graph of the equation.
- Plot the sample points and create a rough sketch of the graph of the equation.
- Test for symmetry. If the equation appears to fail any of the symmetry tests, find a point on the graph of the equation whose reflection fails to be on the graph.
- \(\ y=x^{2}+1\)
- \(\ y=x^{2}-2 x-8\)
- \(\ y=x^{3}-x\)
- \(\ y=\frac{x^{3}}{4}-3 x\)
- \(\ y=\sqrt{x-2}\)
- \(\ y=2 \sqrt{x+4}-2\)
- [MOM] \(\ 3 x-y=7\)
- [MOM] \(\ 3 x-2 y=10\)
- \(\ (x+2)^{2}+y^{2}=16\)
- \(\ x^{2}-y^{2}=1\)
- \(\ 4 y^{2}-9 x^{2}=36\)
- [MOM] \(\ x^{3} y=-4\)
The procedures which we have outlined in the Examples of this section and used in Exercises 41 - 52 all rely on the fact that the equations were “well-behaved”. Not everything in Mathematics is quite so tame, as the following equations will show you. Discuss with your classmates how you might approach graphing the equations given in Exercises 53 - 56. What difficulties arise when trying to apply the various tests and procedures given in this section? For more information, including pictures of the curves, each curve name is a link to its page at www.Wikipedia.org. For a much longer list of fascinating curves, click here.
- \(\ x^{3}+y^{3}-3 x y=0\) Folium of Descartes
- \(\ x^{4}=x^{2}+y^{2}\) Kampyle of Eudoxus
- \(\ y^{2}=x^{3}+3 x^{2}\) Tschirnhausen cubic
- \(\ \left(x^{2}+y^{2}\right)^{2}=x^{3}+y^{3}\) Crooked egg
- With the help of your classmates, find examples of equations whose graphs possess
- symmetry about the x-axis only
- symmetry about the y-axis only
- symmetry about the origin only
- symmetry about the x-axis, y-axis, and origin
Can you find an example of an equation whose graph possesses exactly two of the symmetries listed above? Why or why not?
Answers
- \(\ A=\{(-4,-1),(-2,1),(0,3),(1,4)\}\)
- \(\ C=\{(2, y) \mid y>-3\}\)
- \(\ D=\{(-2, y) \mid-4 \leq y<3\}\)
- \(\ E=\{(x, 2) \mid-4 \leq x<3\}\)
- \(\ F=\{(x, y) \mid y \geq 0\}\)
- \(\ G=\{(x, y) \mid x>-2\}\)
- \(\ H=\{(x, y) \mid-3<x \leq 2\}\)
- \(\ I=\{(x, y) \mid x \geq 0, y \geq 0\}\)
- \(\ J=\{(x, y) \mid-4<x<5,-3<y<2\}\)
- \(\ y=x^{2}+1\)
The graph has no x-intercepts
y-intercept: (0, 1)
\(\ x\) \(\ y\) \(\ (x, y)\) −2 5 (−2, 5) −1 2 (−1, 2) 0 1 (0, 1) 1 2 (1, 2) 2 5 (2, 5) 
The graph is not symmetric about the x-axis (e.g. (2, 5) is on the graph but (2, −5) is not)
The graph is symmetric about the y-axis
The graph is not symmetric about the origin (e.g. (2, 5) is on the graph but (−2, −5) is not)
- \(\ y=x^{2}-2 x-8\)
x-intercepts: (4, 0), (−2, 0)
y-intercept: (0, −8)
\(\ x\) \(\ y\) \(\ (x, y)\) −3 7 (−3, 7) −2 0 (−2, 0) −1 −5 (−1, −5) 0 −8 (0, −8) 1 −9 (1, −9) 2 −8 (2, −8) 3 −5 (3, −5) 4 0 (4, 0) 5 7 (5, 7) 
The graph is not symmetric about the x-axis (e.g. (−3, 7) is on the graph but (−3, −7) is not)
The graph is not symmetric about the y-axis (e.g. (−3, 7) is on the graph but (3, 7) is not)
The graph is not symmetric about the origin (e.g. (−3, 7) is on the graph but (3, −7) is not)
- \(\ y=x^{3}-x\)
x-intercepts: (−1, 0),(0, 0),(1, 0)
y-intercept: (0, 0)
\(\ x\) \(\ y\) \(\ (x, y)\) −2 −6 (−2, −6) −1 0 (−1, 0) 0 0 (0, 0) 1 0 (1, 0) 2 6 (2, 6) 
The graph is not symmetric about the x-axis. (e.g. (2, 6) is on the graph but (2, −6) is not)
The graph is not symmetric about the y-axis. (e.g. (2, 6) is on the graph but (−2, 6) is not)
The graph is symmetric about the origin.
- \(\ y=\frac{x^{3}}{4}-3 x\)
x-intercepts: \(\ (\pm 2 \sqrt{3}, 0),(0,0)\)
y-intercept: \(\ (0, 0)\)
\(\ x\) \(\ y\) \(\ (x, y)\) −4 −4 \(\ (-4,-4)\) −3 \(\ \frac{9}{4}\) \(\ \left(-3, \frac{9}{4}\right)\) −2 4 (−2, 4) −1 \(\ \frac{11}{4}\) \(\ \left(-1, \frac{11}{4}\right)\) 0 0 (0, 0) 1 \(\ -\frac{11}{4}\) \(\ \left(1,-\frac{11}{4}\right)\) 2 −4 (2,-4) 3 \(\ -\frac{9}{4}\) \(\ \left(3,-\frac{9}{4}\right)\) 4 4 (4, 4) 
The graph is not symmetric about the x-axis (e.g. (−4, −4) is on the graph but (−4, 4) is not)
The graph is not symmetric about the y-axis (e.g. (−4, −4) is on the graph but (4, −4) is not)
The graph is symmetric about the origin
- \(\ y=\sqrt{x-2}\)
x-intercept: (2, 0)
The graph has no y-intercepts
\(\ x\) \(\ y\) \(\ (x, y)\) 2 0 (2, 0) 3 1 (3, 1) 6 2 (6, 2) 11 3 (11, 3) 
The graph is not symmetric about the x-axis (e.g. (3, 1) is on the graph but (3, −1) is not)
The graph is not symmetric about the y-axis (e.g. (3, 1) is on the graph but (−3, 1) is not)
The graph is not symmetric about the origin (e.g. (3, 1) is on the graph but (−3, −1) is not)
- \(\ y=2 \sqrt{x+4}-2\)
x-intercept: (−3, 0)
y-intercept: (0, 2)
\(\ x\) \(\ y\) \(\ (x, y)\) −4 −2 (−4, −2) −3 0 (−3, 0) −2 \(\ 2 \sqrt{2}-2\) \(\ (-2, \sqrt{2}-2)\) −1 \(\ 2 \sqrt{3}-2\) \(\ (-2, \sqrt{3}-2)\) 0 2 (0, 2) 1 \(\ 2 \sqrt{5}-2\) \(\ (-2, \sqrt{5}-2) 
The graph is not symmetric about the x-axis (e.g. (−4, −2) is on the graph but (−4, 2) is not)
The graph is not symmetric about the y-axis (e.g. (−4, −2) is on the graph but (4, −2) is not)
The graph is not symmetric about the origin (e.g. (−4, −2) is on the graph but (4, 2) is not)
- \(\ (x+2)^{2}+y^{2}=16\)
Re-write as \(\ y=\pm \sqrt{16-(x+2)^{2}}\).
x-intercepts: \(\ (−6, 0), (2, 0)\)
y-intercepts: \(\ (0, \pm 2 \sqrt{3})\)
\(\ x\) \(\ y\) \(\ (x, y)\) −6 0 (−6, 0) −4 \(\ \pm 2 \sqrt{3}\) \(\ (-4, \pm 2 \sqrt{3})\) −2 \(\ \pm 4\) \(\ (-2, \pm 4)\) 0 \(\ \pm 2 \sqrt{3}\) \(\ (0, \pm 2 \sqrt{3})\) 2 0 (2, 0) 
The graph is symmetric about the x-axis
The graph is not symmetric about the y-axis (e.g. (−6, 0) is on the graph but (6, 0) is not)
The graph is not symmetric about the origin (e.g. (−6, 0) is on the graph but (6, 0) is not)
- \(\ x^{2}-y^{2}=1\)
Re-write as: \(\ y=\pm \sqrt{x^{2}-1}\).
x-intercep: (−1, 0), (1, 0)
The graph has no y-intercepts
\(\ x\) \(\ y\) \(\ (x, y)\) −3 \(\ \pm \sqrt{8}\) \(\ (-3, \pm \sqrt{8})\) −2 \(\ \pm \sqrt{3}\) \(\ (-2, \pm \sqrt{3})\) −1 0 (−1, 0) 1 0 (1, 0) 2 \(\ \pm \sqrt{3}\) \(\ (2, \pm \sqrt{3})\) 3 \(\ \pm \sqrt{8}\) \(\ (3, \pm \sqrt{8})\) 
The graph is symmetric about the x-axis
The graph is symmetric about the y-axis
The graph is symmetric about the origin
- \(\ 4 y^{2}-9 x^{2}=36\)
Re-write as: \(\ y=\pm \frac{\sqrt{9 x^{2}+36}}{2}\).
The graph has no x-intercepts
y-intercepts: \(\ (0, \pm 3)\)
\(\ x\) \(\ y\) \(\ (x, y)\) −4 \(\ \pm 3 \sqrt{5}\) \(\ (-4, \pm 3 \sqrt{5})\) −2 \(\ \pm 3 \sqrt{2}\) \(\ (-2, \pm 3 \sqrt{2})\) 0 \(\ \pm 3\) \(\ (0, \pm 3)\) 2 \(\ \pm 3 \sqrt{2}\) \(\ (2, \pm 3 \sqrt{2})\) 4 \(\ \pm 3 \sqrt{5}\) \(\ (4, \pm 3 \sqrt{5})\) 
The graph is symmetric about the x-axis
The graph is symmetric about the y-axis
The graph is symmetric about the origin