Warm-up and HW Quiz
Warm-up and HW Quiz
Warm-up and HW Quiz
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SECTION 9.1<br />
WARM-UP EXERCISES<br />
Sketch the graph of the quadratic function. Identify the vertex, axis of symmetry, <strong>and</strong><br />
x-intercept(s).<br />
1. y = 3(x + 1)2 − 3 2. y = x 2 + 6x + 5<br />
3. y = 1 3 x 2 + 2 4.<br />
y = 36 − 9x<br />
2<br />
DAILY HOMEWORK QUIZ<br />
Find the vertex, focus, <strong>and</strong> directrix of the parabola <strong>and</strong> sketch its graph.<br />
1. x 2 = −4y<br />
2. (x − 3) − 4(y + 2)2 = 0<br />
Find the st<strong>and</strong>ard form of the equation of the parabola with the given characteristics.<br />
3. Vertex: (1, –2); focus (1, –4)<br />
4. Vertex: (0, 3); directrix: y = 0<br />
Identify the center <strong>and</strong> radius of the circle.<br />
5. (x + 1)2 + (y − 2) 2 = 25<br />
6. x 2 + 2x + y 2 + 2y + 1= 0<br />
ANSWERS<br />
WU 1. 2.<br />
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3. 4.<br />
QUIZ 1. vertex: (0, 0); focus: (0, –1); directrix: y = 1<br />
2. vertex: (3, –2); focus:<br />
⎛ 49<br />
⎝<br />
⎜<br />
16 , −2 ⎞<br />
⎠<br />
⎟<br />
; directrix: x = −<br />
47<br />
16<br />
3. (x − 1)2 = −8(y + 2) 4. x 2 = 8(y − 3)<br />
5. Center: (–1, 2); Radius: 5 6. Center: (–1, –1); Radius: 1<br />
82<br />
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SECTION 9.2<br />
WARM-UP EXERCISES<br />
Find the st<strong>and</strong>ard form of the equation of the parabola with the given characteristics.<br />
1. Vertex: (3, 2); focus (0, 2)<br />
2. Vertex: (–1, 5); directrix: y = 2<br />
3. Focus: (0, 1); directrix: x = –1<br />
Find the vertex, focus, <strong>and</strong> directrix of the parabola <strong>and</strong> sketch its graph.<br />
4. y 2 = 4x<br />
5. (x − 2)2 + 8(y + 1) = 0<br />
DAILY HOMEWORK QUIZ<br />
Find the center, vertices, foci, <strong>and</strong> eccentricity of the ellipse.<br />
1. x 2<br />
4 + y 2<br />
3 = 1<br />
(y −<br />
2. (x + 2) 2 3)2<br />
+<br />
1/ 8<br />
= 1<br />
3.<br />
(x − 4) 2<br />
16<br />
+ y 2<br />
4 = 1<br />
Find the st<strong>and</strong>ard form of the equation of the ellipse with the given characteristics.<br />
4. Vertices: (±3, 0) ; foci: (±1, 0)<br />
5. Center: (−1, −2) ; vertex: (3, −2) ; major axis of length 10<br />
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83
ANSWERS<br />
WU 1. (y − 2)2 = −12(x − 3)<br />
2. (x − 5)2 = 12(y + 1)<br />
3. (y − 1)2 = 4(x − 1)<br />
4. vertex: (0, 0); focus: (1, 0); directrix: x = –1<br />
5. vertex: (2, –1); focus: (2, –3); directrix: y = 1<br />
QUIZ 1. center: (0, 0); vertices: (±2, 0) ; foci: (±1, 0); e ≈ 0.5<br />
2. center: (–2, 3); vertices: (–3, 3) <strong>and</strong> (–1, 3); foci: (−2 ± 63<br />
8 , 3);<br />
e ≈ 0.9922<br />
3. center: (4, 0); vertices: (8, 0) <strong>and</strong> (0, 0); foci: (4 ± 2 3, 0) ; e ≈ 0.866<br />
4.<br />
x 2<br />
9 + y 2<br />
8 = 1<br />
5.<br />
(x + 1) 2<br />
16<br />
(y + 2)2<br />
+ = 1<br />
25<br />
84<br />
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SECTION 9.3<br />
WARM-UP EXERCISES<br />
Identify the conic as a circle or an ellipse. Then find the center, radius, <strong>and</strong> foci of the<br />
conic (if applicable).<br />
1.<br />
x 2<br />
25 + y 2<br />
8 = 1 2. (x − 2) 2<br />
16 / 9<br />
+<br />
(y + 2)2<br />
16 / 9 = 1<br />
3. x 2 + y 2 − 4x + 12y − 10 = 0 4. 5x 2 + 4y 2 − 20x − 2y + 4 = 0<br />
DAILY HOMEWORK QUIZ<br />
Find the center, vertices, foci, <strong>and</strong> asymptotes of the hyperbola.<br />
1. x 2 − y 2<br />
4 = 1 2. (x − 2) 2<br />
1/ 9<br />
−<br />
(y + 2)2<br />
1/ 16 = 1 3. 8y 2 − x 2 + 2x + 32y − 47 = 0<br />
Find the st<strong>and</strong>ard form of the equation of the hyperbola with the given characteristics<br />
<strong>and</strong> center at the origin.<br />
4. Vertices: ±2, 0<br />
ANSWERS<br />
( ) ; foci: ( ±4, 0) 5. Vertices: ( 0, ±8) ; asymptotes: y = ±x<br />
WU 1. ellipse; center: (0, 0); foci: (0, ± 17)<br />
2. circle; center: (2, –2); radius: 4 3<br />
3. circle; center: (2, –6); radius: 5 2<br />
4. ellipse; center: (2, 1); foci: (2, 0) <strong>and</strong> (2, 2)<br />
QUIZ 1. center: (0, 0); vertices: (±1, 0); foci: (± 5, 0); asymptotes: y = ±2x<br />
2. center: (2, –2); vertices:(± 1 5<br />
, 0) ; foci: (±<br />
3 12 , 0) ; asymptotes: y = ± 3 4 x<br />
3. center: (–2, 1); vertices: (−2, 1± 2) ; foci: (−2, 1± 3 2) ;<br />
4.<br />
asymptotes: y = −2 ± 2<br />
4<br />
x 2<br />
4 − y 2<br />
12 = 1 5. y 2<br />
(x − 1)<br />
64 − x 2<br />
64 = 1<br />
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SECTION 9.4<br />
WARM-UP EXERCISES<br />
Classify the graph of each equation as a circle, a parabola, an ellipse, or a hyperbola.<br />
1. x 2 + y 2 − 2x + 24y − 6 = 0<br />
2. 3y 2 + 2x + 9y − 7 = 0<br />
3. 8y 2 − 4x 2 − 16y + 26 = 0<br />
Given the value of cot θ, find the value of cos θ <strong>and</strong> sin θ.<br />
4. cotθ = 2<br />
5. cotθ = 2 5<br />
6. cotθ = 4 3<br />
DAILY HOMEWORK QUIZ<br />
The x’y’-coordinate system has been rotated θ degrees from the xy-coordinate<br />
system. The coordinates of a point in the xy-coordinate system are given. Find the<br />
coordinates of the point in the rotated coordinate system.<br />
1.<br />
2.<br />
θ = 90°, (−2, 0)<br />
θ = 60°, (1, 3)<br />
Rotate the axes to eliminate the xy-term in the equation. Then write the equation in<br />
st<strong>and</strong>ard form. Sketch the graph of the resulting equation, showing both sets of axes.<br />
3. 7x 2 − 6 3xy + 13y 2 − 16 = 0<br />
4. x 2 − 2xy + y 2 − x − y = 0<br />
86<br />
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ANSWERS<br />
WU 1. circle 2. parabola 3. hyperbola<br />
4. cosθ = 2 ; sinθ = 1<br />
5 5<br />
5. cosθ = 2 ; sinθ = 5<br />
29 29<br />
6. cosθ = 4 5 ; sinθ = 3 5<br />
QUIZ 1. (0, –2)<br />
2. approximately (–2.10, 2.37)<br />
x′ 2<br />
3.<br />
4 + y′2 = 1;<br />
4. x′ = 2y′2 ;<br />
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87
SECTION 9.5<br />
WARM-UP EXERCISES<br />
Evaluate the function for the given values of t.<br />
1. y = 2t 2 + 1; t = 0, 1, 2, 3, <strong>and</strong> 4<br />
2. y = 4 − 2t ; t = 0, 1, 2, 3, <strong>and</strong> 4<br />
3. y = 2cost; t = 0, π 3 , 2π 3 , π<br />
Sketch a graph of the following functions for positive x-values.<br />
4. y 2 + 2x − 4y = 0 5. (x − 1)2 + (y − 1) 2 = 4<br />
DAILY HOMEWORK QUIZ<br />
Sketch the curve represented by the parametric equations (indicate the orientation of<br />
the curve). Then eliminate the parameter <strong>and</strong> write the corresponding rectangular<br />
equation whose graph represents the curve.<br />
1. x = 3t − 2<br />
2. x = 2t<br />
3. x = e −t<br />
4. x = 2 − cosθ<br />
y = 2t + 1<br />
y = 1<br />
1− t<br />
y = e 2t<br />
y = 1+ 2sinθ<br />
88<br />
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ANSWERS<br />
WU 1. 1, 3, 9, 19, 33 2. 4, 2, 0, 2, 4 3. 2, 1, –1, –2<br />
4. 5.<br />
QUIZ 1. 2.<br />
y = 2x + 7<br />
3<br />
3. 4.<br />
y = 2<br />
2 − x<br />
y = 1<br />
(y −<br />
(x − 2) 2 1)2<br />
+<br />
x 2 4<br />
= 1<br />
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SECTION 9.6<br />
WARM-UP EXERCISES<br />
Find the slope of the line with inclination θ.<br />
1. θ = 2.2 radians 2. θ = 1 radian<br />
Eliminate the parameter <strong>and</strong> write the corresponding rectangular equation.<br />
3. x = t − 2<br />
y = 2 − t<br />
4. x = cosθ<br />
y = 2sinθ<br />
5. x = 2 + 3cosθ<br />
y = 4 + 2sinθ<br />
DAILY HOMEWORK QUIZ<br />
A point in polar coordinates is given. Find the corresponding rectangular coordinates.<br />
1. (2, −π ) 2. ⎛<br />
−2, 3π ⎞<br />
⎝<br />
⎜<br />
4 ⎠<br />
⎟<br />
Convert the rectangular equation to polar form. Assume a > 0.<br />
3. x 2 + y 2 = 4a 2<br />
4. y = 2a<br />
5. xy = −4a<br />
ANSWERS<br />
WU 1. about –1.37 2. about 1.56 3. x + y = 0<br />
4. x 2 + y 2<br />
4 = 1 5. (x − 2) 2<br />
9<br />
(y − 4)2<br />
+ = 1<br />
4<br />
QUIZ 1.<br />
(−2, 0) 2. ( 2, − 2) 3. r = 2a<br />
4. r = 2a<br />
sinθ<br />
4a<br />
5. r 2 =<br />
cosθ sinθ<br />
90<br />
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SECTION 9.7<br />
WARM-UP EXERCISES<br />
A point in rectangular coordinates is given. Find the corresponding polar coordinates.<br />
1. (2, 0)<br />
2. (− 2, 2)<br />
Convert the polar equation to rectangular form.<br />
3. r = 5<br />
4. r = −2cscθ<br />
5. θ = 7π 6<br />
DAILY HOMEWORK QUIZ<br />
Test for symmetry with respect to θ = !/2, the polar axis, <strong>and</strong> the pole.<br />
1. r = 1+ 2cosθ<br />
2. r 2 = 4cos2θ<br />
Sketch the graph of the polar equation.<br />
3. r = 4<br />
4. r = 3sinθ<br />
2<br />
5. r 2 = 4cos2θ<br />
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91
ANSWERS<br />
WU 1. (2, 0) 2. ⎛<br />
2, 3π ⎝<br />
⎜<br />
4<br />
⎞<br />
⎠<br />
⎟ 3. x 2 + y 2 = 25<br />
QUIZ<br />
4.<br />
y = −2 5. y =<br />
3x<br />
3<br />
1. symmetry with respect to the polar axis<br />
2. symmetry with respect to the polar axis <strong>and</strong> θ = !/2<br />
3.<br />
4.<br />
5.<br />
92<br />
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SECTION 9.8<br />
WARM-UP EXERCISES<br />
Sketch the graph of the polar equation.<br />
1. r = 4 − 3sinθ 2. r = 2sin2θ<br />
3. r =<br />
3<br />
2cosθ − sinθ<br />
4. r 2 = 9sinθ<br />
DAILY HOMEWORK QUIZ<br />
Identify the conic <strong>and</strong> sketch its graph.<br />
3<br />
1. r =<br />
1− sinθ<br />
2. r =<br />
4<br />
2 − 3sinθ<br />
5<br />
3. r =<br />
2 − cosθ<br />
Find a polar equation of a conic with the given eccentricity <strong>and</strong> directrix, <strong>and</strong> its focus<br />
at the pole.<br />
4. parabola: e = 1, x = –2 5. ellipse: e = 1 2 , y = –3<br />
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ANSWERS<br />
WU 1. 2.<br />
3. 4.<br />
QUIZ 1. a parabola; 2. a hyperbola;<br />
3. an ellipse;<br />
2<br />
4. r =<br />
1− cosθ<br />
3<br />
5. r =<br />
2 − sinθ<br />
94<br />
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