Stewart Precalc 6e Chapter 11 Review: Concept Check

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Stewart Precalc 6e Chapter 11 Review: Concept Check 0/74
1 Parabolas · Level 2
Find the vertex, focus, and directrix of the parabola \(y^2 = 4 x\), and sketch the graph.
2 Parabolas · Level 2
Find the vertex, focus, and directrix of the parabola \(x = \dfrac{1}{12} y^2\), and sketch the graph.
3 Parabolas · Level 2
Find the vertex, focus, and directrix of the parabola \(x^2 + 8 y = 0\), and sketch the graph.
4 Parabolas · Level 2
Find the vertex, focus, and directrix of the parabola \(2 x - y^2 = 0\), and sketch the graph.
5 Parabolas · Level 3
Find the vertex, focus, and directrix of the parabola \(x - y^2 + 4 y - 2 = 0\), and sketch the graph.
6 Parabolas · Level 3
Find the vertex, focus, and directrix of the parabola \(2 x^2 + 6 x + 5 y + 10 = 0\), and sketch the graph.
7 Parabolas · Level 3
Find the vertex, focus, and directrix of the parabola \(\dfrac{1}{2} x^2 + 2 x = 2 y + 4\), and sketch the graph.
8 Parabolas · Level 3
Find the vertex, focus, and directrix of the parabola \(x^2 = 3(x + y)\), and sketch the graph.
9 Ellipses · Level 2
Find the center, vertices, foci, and the lengths of the major and minor axes of the ellipse \(\dfrac{x^2}{9} + \dfrac{y^2}{25} = 1\), and sketch the graph.
10 Ellipses · Level 2
Find the center, vertices, foci, and the lengths of the major and minor axes of the ellipse \(\dfrac{x^2}{49} + \dfrac{y^2}{9} = 1\), and sketch the graph.
11 Ellipses · Level 2
Find the center, vertices, foci, and the lengths of the major and minor axes of the ellipse \(x^2 + 4 y^2 = 16\), and sketch the graph.
12 Ellipses · Level 2
Find the center, vertices, foci, and the lengths of the major and minor axes of the ellipse \(9 x^2 + 4 y^2 = 1\), and sketch the graph.
13 Ellipses · Level 2
Find the center, vertices, foci, and the lengths of the major and minor axes of the ellipse \(\dfrac{(x - 3)^2}{9} + \dfrac{y^2}{16} = 1\), and sketch the graph.
14 Ellipses · Level 2
Find the center, vertices, foci, and the lengths of the major and minor axes of the ellipse \(\dfrac{(x - 2)^2}{25} + \dfrac{(y + 3)^2}{16} = 1\), and sketch the graph.
15 Ellipses · Level 3
Find the center, vertices, foci, and the lengths of the major and minor axes of the ellipse \(4 x^2 + 9 y^2 = 36 y\), and sketch the graph.
16 Ellipses · Level 3
Find the center, vertices, foci, and the lengths of the major and minor axes of the ellipse \(2 x^2 + y^2 = 2 + 4(x - y)\), and sketch the graph.
17 Hyperbolas · Level 2
Find the center, vertices, foci, and asymptotes of the hyperbola \(-\dfrac{x^2}{9} + \dfrac{y^2}{16} = 1\), and sketch the graph.
18 Hyperbolas · Level 2
Find the center, vertices, foci, and asymptotes of the hyperbola \(\dfrac{x^2}{49} - \dfrac{y^2}{32} = 1\), and sketch the graph.
19 Hyperbolas · Level 2
Find the center, vertices, foci, and asymptotes of the hyperbola \(x^2 - 2 y^2 = 16\), and sketch the graph.
20 Hyperbolas · Level 2
Find the center, vertices, foci, and asymptotes of the hyperbola \(x^2 - 4 y^2 + 16 = 0\), and sketch the graph.
21 Hyperbolas · Level 2
Find the center, vertices, foci, and asymptotes of the hyperbola \(\dfrac{(x + 4)^2}{16} - \dfrac{y^2}{16} = 1\), and sketch the graph.
22 Hyperbolas · Level 2
Find the center, vertices, foci, and asymptotes of the hyperbola \(\dfrac{(x - 2)^2}{8} - \dfrac{(y + 2)^2}{8} = 1\), and sketch the graph.
23 Hyperbolas · Level 3
Find the center, vertices, foci, and asymptotes of the hyperbola \(9 y^2 + 18 y = x^2 + 6 x + 18\), and sketch the graph.
24 Hyperbolas · Level 3
Find the center, vertices, foci, and asymptotes of the hyperbola \(y^2 = x^2 + 6 y\), and sketch the graph.
25 Identify Conics from Graph · Level 3
Find an equation for the conic whose graph is shown.
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26 Identify Conics from Graph · Level 3
Find an equation for the conic whose graph is shown.
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27 Identify Conics from Graph · Level 3
Find an equation for the conic whose graph is shown.
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28 Identify Conics from Graph · Level 3
Find an equation for the conic whose graph is shown.
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29 Identify Conics from Graph · Level 3
Find an equation for the conic whose graph is shown.
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30 Identify Conics from Graph · Level 3
Find an equation for the conic whose graph is shown.
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31 Identify and Graph · Level 3
Determine the type of curve represented by the equation \(\dfrac{x^2}{12} + y = 1\). Find the foci and vertices (if any), and sketch the graph.
32 Identify and Graph · Level 3
Determine the type of curve represented by the equation \(\dfrac{x^2}{12} + \dfrac{y^2}{144} = \dfrac{y}{12}\). Find the foci and vertices (if any), and sketch the graph.
33 Identify and Graph · Level 2
Determine the type of curve represented by the equation \(x^2 - y^2 + 144 = 0\). Find the foci and vertices (if any), and sketch the graph.
34 Identify and Graph · Level 3
Determine the type of curve represented by the equation \(x^2 + 6 x = 9 y^2\). Find the foci and vertices (if any), and sketch the graph.
35 Identify and Graph · Level 3
Determine the type of curve represented by the equation \(4 x^2 + y^2 = 8(x + y)\). Find the foci and vertices (if any), and sketch the graph.
36 Identify and Graph · Level 3
Determine the type of curve represented by the equation \(3 x^2 - 6(x + y) = 10\). Find the foci and vertices (if any), and sketch the graph.
37 Identify and Graph · Level 3
Determine the type of curve represented by the equation \(x = y^2 - 16 y\). Find the foci and vertices (if any), and sketch the graph.
38 Identify and Graph · Level 3
Determine the type of curve represented by the equation \(2 x^2 + 4 = 4 x + y^2\). Find the foci and vertices (if any), and sketch the graph.
39 Identify and Graph · Level 3
Determine the type of curve represented by the equation \(2 x^2 - 12 x + y^2 + 6 y + 26 = 0\). Find the foci and vertices (if any), and sketch the graph.
40 Identify and Graph · Level 3
Determine the type of curve represented by the equation \(36 x^2 - 4 y^2 - 36 x - 8 y = 31\). Find the foci and vertices (if any), and sketch the graph.
41 Identify and Graph · Level 3
Determine the type of curve represented by the equation \(9 x^2 + 8 y^2 - 15 x + 8 y + 27 = 0\). Find the foci and vertices (if any), and sketch the graph.
42 Identify and Graph · Level 2
Determine the type of curve represented by the equation \(x^2 + 4 y^2 = 4 x + 8\). Find the foci and vertices (if any), and sketch the graph.
43 Equation from Properties · Level 2
Find an equation for the parabola with focus \(F(0, 1)\) and directrix \(y = -1\).
44 Equation from Properties · Level 3
Find an equation for the ellipse with center \(C(0, 4)\), foci \(F_1(0, 0)\) and \(F_2(0, 8)\), and major axis of length 10.
45 Equation from Properties · Level 3
Find an equation for the hyperbola with vertices \(V(0, \pm 2)\) and asymptotes \(y = \pm \dfrac{1}{2} x\).
46 Equation from Properties · Level 3
Find an equation for the hyperbola with center \(C(2, 4)\), foci \(F_1(2, 1)\) and \(F_2(2, 7)\), and vertices \(V_1(2, 6)\) and \(V_2(2, 2)\).
47 Equation from Properties · Level 4
Find an equation for the ellipse with foci \(F_1(1, 1)\) and \(F_2(1, 3)\), and with one vertex on the x-axis.
48 Equation from Properties · Level 2
Find an equation for the parabola with vertex \(V(5, 5)\) and directrix the y-axis.
49 Equation from Properties · Level 4
Find an equation for the ellipse with vertices \(V_1(7, 12)\) and \(V_2(7, -8)\), and passing through the point \(P(1, 8)\).
50 Equation from Properties · Level 3
Find an equation for the parabola with vertex \(V(-1, 0)\) and horizontal axis of symmetry, and crossing the y-axis at \(y = 2\).
51 Applications · Level 3
The path of the earth around the sun is an ellipse with the sun at one focus. The ellipse has major axis 186,000,000 mi and eccentricity 0.017. Find the distance between the earth and the sun when the earth is (a) closest to the sun and (b) farthest from the sun.
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52 Applications · Level 4
A ship is located 40 mi from a straight shoreline. LORAN stations A and B are located on the shoreline, 300 mi apart. From the LORAN signals, the captain determines that his ship is 80 mi closer to A than to B. Find the location of the ship. (Place A and B on the y-axis with the x-axis halfway between them. Find the x- and y-coordinates of the ship.)
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53 Families of Conics · Level 4
(a) Draw graphs of the following family of ellipses for \(k = 1, 2, 4, 8\). \(\dfrac{x^2}{16 + k^2} + \dfrac{y^2}{k^2} = 1\)
(b) Prove that all the ellipses in part (a) have the same foci.

Enter your answer directly below each part above.

54 Families of Conics · Level 3
(a) Draw graphs of the following family of parabolas for \(k = \dfrac{1}{2}, 1, 2, 4\). \(y = k x^2\)
(b) Find the foci of the parabolas in part (a).
(c) How does the location of the focus change as \(k\) increases?

Enter your answer directly below each part above.

55 Rotation of Axes · Level 4
An equation of a conic is given. (a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\)-term. (c) Sketch the graph. \(x^2 + 4 x y + y^2 = 1\)
56 Rotation of Axes · Level 4
An equation of a conic is given. (a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\)-term. (c) Sketch the graph. \(5 x^2 - 6 x y + 5 y^2 - 8 x + 8 y - 8 = 0\)
57 Rotation of Axes · Level 5
An equation of a conic is given. (a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\)-term. (c) Sketch the graph. \(7 x^2 - 6 \sqrt{3} x y + 13 y^2 - 4 \sqrt{3} x - 4 y = 0\)
58 Rotation of Axes · Level 4
An equation of a conic is given. (a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\)-term. (c) Sketch the graph. \(9 x^2 + 24 x y + 16 y^2 = 25\)
59 Graphing Device · Level 2
Use a graphing device to graph the conic \(5 x^2 + 3 y^2 = 60\). Identify the type of conic from the graph.
60 Graphing Device · Level 2
Use a graphing device to graph the conic \(9 x^2 - 12 y^2 + 36 = 0\). Identify the type of conic from the graph.
61 Graphing Device · Level 2
Use a graphing device to graph the conic \(6 x + y^2 - 12 y = 30\). Identify the type of conic from the graph.
62 Graphing Device · Level 3
Use a graphing device to graph the conic \(52 x^2 - 72 x y + 73 y^2 = 100\). Identify the type of conic from the graph.
63 Polar Conics · Level 3
A polar equation of a conic is given. (a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. \(r = \dfrac{1}{1 - \cos \theta}\)
64 Polar Conics · Level 3
A polar equation of a conic is given. (a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. \(r = \dfrac{2}{3 + 2 \sin \theta}\)
65 Polar Conics · Level 3
A polar equation of a conic is given. (a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. \(r = \dfrac{4}{1 + 2 \sin \theta}\)
66 Polar Conics · Level 3
A polar equation of a conic is given. (a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. \(r = \dfrac{12}{1 - 4 \cos \theta}\)
67 Concept Review - Hyperbola · Level 2
(a) Give the geometric definition of a hyperbola. What are the foci of the hyperbola?
(b) For the hyperbola with equation \(\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1\), what are the coordinates of the vertices and foci? What are the equations of the asymptotes? What is the transverse axis? Illustrate with a graph.
(c) State the equation of a hyperbola with foci on the y-axis.
(d) What steps would you take to sketch a hyperbola with a given equation?

Enter your answer directly below each part above.

68 Concept Review - Translations · Level 2
Suppose \(h\) and \(k\) are positive numbers. What is the effect on the graph of an equation in \(x\) and \(y\) if
(a) \(x\) is replaced by \(x - h\)? By \(x + h\)?
(b) \(y\) is replaced by \(y - k\)? By \(y + k\)?

Enter your answer directly below each part above.

69 Concept Review - Identifying Conics · Level 2
How can you tell whether the following nondegenerate conic is a parabola, an ellipse, or a hyperbola? \(A x^2 + C y^2 + D x + E y + F = 0\)
70 Concept Review - Rotation · Level 2
Suppose the x- and y-axes are rotated through an acute angle \(\phi\) to produce the X- and Y-axes. Write equations that relate the coordinates \((x, y)\) and \((X, Y)\) of a point in the xy-plane and XY-plane, respectively.
71 Concept Review - Discriminant · Level 3
(a) How do you eliminate the \(x y\)-term in this equation? \(A x^2 + B x y + C y^2 + D x + E y + F = 0\)
(b) What is the discriminant of the conic in part (a)? How can you use the discriminant to determine whether the conic is a parabola, an ellipse, or a hyperbola?

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72 Concept Review - Polar Conics · Level 2
(a) Write polar equations that represent a conic with eccentricity \(e\).
(b) For what values of \(e\) is the conic an ellipse? A hyperbola? A parabola?

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73 Concept Check - Parabola · Level 2
(a) Give the geometric definition of a parabola. What are the focus and directrix of the parabola?
(b) Sketch the parabola \(x^2 = 4 p y\) for the case \(p > 0\). Identify on your diagram the vertex, focus, and directrix. What happens if \(p < 0\)?
(c) Sketch the parabola \(y^2 = 4 p x\), together with its vertex, focus, and directrix, for the case \(p > 0\). What happens if \(p < 0\)?

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74 Concept Check - Ellipse · Level 2
(a) Give the geometric definition of an ellipse. What are the foci of the ellipse?
(b) For the ellipse with equation \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) where \(a > b > 0\), what are the coordinates of the vertices and the foci? What are the major and minor axes? Illustrate with a graph.
(c) Give an expression for the eccentricity of the ellipse in part (b).
(d) State the equation of an ellipse with foci on the \(y\)-axis.

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