Warm-up and HW Quiz

SECTION 9.1 

WARM-UP EXERCISES 

Sketch the graph of the quadratic function. Identify the vertex, axis of symmetry, and 

x-intercept(s). 

1. y = 3(x + 1)2 − 3 2. y = x 2 + 6x + 5 

3. y = 1 3 x 2 + 2 4. 

y = 36 − 9x 

2 

DAILY HOMEWORK QUIZ 

Find the vertex, focus, and directrix of the parabola and sketch its graph. 

1. x 2 = −4y 

2. (x − 3) − 4(y + 2)2 = 0 

Find the standard form of the equation of the parabola with the given characteristics. 

3. Vertex: (1, –2); focus (1, –4) 

4. Vertex: (0, 3); directrix: y = 0 

Identify the center and radius of the circle. 

5. (x + 1)2 + (y − 2) 2 = 25 

6. x 2 + 2x + y 2 + 2y + 1= 0 

ANSWERS 

WU 1. 2. 

Precalculus with Limits: A Graphing Approach 5e, Warm-Up Exercises and Daily Homework Quiz Transparencies 

Copyright © Houghton Mifflin Company. All rights reserved. 

81

3. 4. 

QUIZ 1. vertex: (0, 0); focus: (0, –1); directrix: y = 1 

2. vertex: (3, –2); focus: 

⎛ 49 

⎝ 

⎜ 

16 , −2 ⎞ 

⎠ 

⎟ 

; directrix: x = − 

47 

16 

3. (x − 1)2 = −8(y + 2) 4. x 2 = 8(y − 3) 

5. Center: (–1, 2); Radius: 5 6. Center: (–1, –1); Radius: 1 

82 


Copyright © Houghton Mifflin Company. All rights reserved.

SECTION 9.2 


Find the standard form of the equation of the parabola with the given characteristics. 

1. Vertex: (3, 2); focus (0, 2) 

2. Vertex: (–1, 5); directrix: y = 2 

3. Focus: (0, 1); directrix: x = –1 

Find the vertex, focus, and directrix of the parabola and sketch its graph. 

4. y 2 = 4x 

5. (x − 2)2 + 8(y + 1) = 0 


Find the center, vertices, foci, and eccentricity of the ellipse. 

1. x 2 

4 + y 2 

3 = 1 

(y − 

2. (x + 2) 2 3)2 

+ 

1/ 8 

= 1 

3. 

(x − 4) 2 

16 

+ y 2 

4 = 1 

Find the standard form of the equation of the ellipse with the given characteristics. 

4. Vertices: (±3, 0) ; foci: (±1, 0) 

5. Center: (−1, −2) ; vertex: (3, −2) ; major axis of length 10 



83

ANSWERS 

WU 1. (y − 2)2 = −12(x − 3) 

2. (x − 5)2 = 12(y + 1) 

3. (y − 1)2 = 4(x − 1) 

4. vertex: (0, 0); focus: (1, 0); directrix: x = –1 

5. vertex: (2, –1); focus: (2, –3); directrix: y = 1 

QUIZ 1. center: (0, 0); vertices: (±2, 0) ; foci: (±1, 0); e ≈ 0.5 

2. center: (–2, 3); vertices: (–3, 3) and (–1, 3); foci: (−2 ± 63 

8 , 3); 

e ≈ 0.9922 

3. center: (4, 0); vertices: (8, 0) and (0, 0); foci: (4 ± 2 3, 0) ; e ≈ 0.866 

4. 

x 2 

9 + y 2 

8 = 1 

5. 

(x + 1) 2 

16 

(y + 2)2 

+ = 1 

25 

84 



SECTION 9.3 


Identify the conic as a circle or an ellipse. Then find the center, radius, and foci of the 

conic (if applicable). 

1. 

x 2 

25 + y 2 

8 = 1 2. (x − 2) 2 

16 / 9 

+ 

(y + 2)2 

16 / 9 = 1 

3. x 2 + y 2 − 4x + 12y − 10 = 0 4. 5x 2 + 4y 2 − 20x − 2y + 4 = 0 


Find the center, vertices, foci, and asymptotes of the hyperbola. 

1. x 2 − y 2 

4 = 1 2. (x − 2) 2 

1/ 9 

− 

(y + 2)2 

1/ 16 = 1 3. 8y 2 − x 2 + 2x + 32y − 47 = 0 

Find the standard form of the equation of the hyperbola with the given characteristics 

and center at the origin. 

4. Vertices: ±2, 0 

ANSWERS 

( ) ; foci: ( ±4, 0) 5. Vertices: ( 0, ±8) ; asymptotes: y = ±x 

WU 1. ellipse; center: (0, 0); foci: (0, ± 17) 

2. circle; center: (2, –2); radius: 4 3 

3. circle; center: (2, –6); radius: 5 2 

4. ellipse; center: (2, 1); foci: (2, 0) and (2, 2) 

QUIZ 1. center: (0, 0); vertices: (±1, 0); foci: (± 5, 0); asymptotes: y = ±2x 

2. center: (2, –2); vertices:(± 1 5 

, 0) ; foci: (± 

3 12 , 0) ; asymptotes: y = ± 3 4 x 

3. center: (–2, 1); vertices: (−2, 1± 2) ; foci: (−2, 1± 3 2) ; 

4. 

asymptotes: y = −2 ± 2 

4 

x 2 

4 − y 2 

12 = 1 5. y 2 

(x − 1) 

64 − x 2 

64 = 1 



85

SECTION 9.4 


Classify the graph of each equation as a circle, a parabola, an ellipse, or a hyperbola. 

1. x 2 + y 2 − 2x + 24y − 6 = 0 

2. 3y 2 + 2x + 9y − 7 = 0 

3. 8y 2 − 4x 2 − 16y + 26 = 0 

Given the value of cot θ, find the value of cos θ and sin θ. 

4. cotθ = 2 

5. cotθ = 2 5 

6. cotθ = 4 3 


The x’y’-coordinate system has been rotated θ degrees from the xy-coordinate 

system. The coordinates of a point in the xy-coordinate system are given. Find the 

coordinates of the point in the rotated coordinate system. 

1. 

2. 

θ = 90°, (−2, 0) 

θ = 60°, (1, 3) 

Rotate the axes to eliminate the xy-term in the equation. Then write the equation in 

standard form. Sketch the graph of the resulting equation, showing both sets of axes. 

3. 7x 2 − 6 3xy + 13y 2 − 16 = 0 

4. x 2 − 2xy + y 2 − x − y = 0 

86 



ANSWERS 

WU 1. circle 2. parabola 3. hyperbola 

4. cosθ = 2 ; sinθ = 1 

5 5 

5. cosθ = 2 ; sinθ = 5 

29 29 

6. cosθ = 4 5 ; sinθ = 3 5 

QUIZ 1. (0, –2) 

2. approximately (–2.10, 2.37) 

x′ 2 

3. 

4 + y′2 = 1; 

4. x′ = 2y′2 ; 



87

SECTION 9.5 


Evaluate the function for the given values of t. 

1. y = 2t 2 + 1; t = 0, 1, 2, 3, and 4 

2. y = 4 − 2t ; t = 0, 1, 2, 3, and 4 

3. y = 2cost; t = 0, π 3 , 2π 3 , π 

Sketch a graph of the following functions for positive x-values. 

4. y 2 + 2x − 4y = 0 5. (x − 1)2 + (y − 1) 2 = 4 


Sketch the curve represented by the parametric equations (indicate the orientation of 

the curve). Then eliminate the parameter and write the corresponding rectangular 

equation whose graph represents the curve. 

1. x = 3t − 2 

2. x = 2t 

3. x = e −t 

4. x = 2 − cosθ 

y = 2t + 1 

y = 1 

1− t 

y = e 2t 

y = 1+ 2sinθ 

88 



ANSWERS 

WU 1. 1, 3, 9, 19, 33 2. 4, 2, 0, 2, 4 3. 2, 1, –1, –2 

4. 5. 

QUIZ 1. 2. 

y = 2x + 7 

3 

3. 4. 

y = 2 

2 − x 

y = 1 

(y − 

(x − 2) 2 1)2 

+ 

x 2 4 

= 1 



89

SECTION 9.6 


Find the slope of the line with inclination θ. 

1. θ = 2.2 radians 2. θ = 1 radian 

Eliminate the parameter and write the corresponding rectangular equation. 

3. x = t − 2 

y = 2 − t 

4. x = cosθ 

y = 2sinθ 

5. x = 2 + 3cosθ 

y = 4 + 2sinθ 


A point in polar coordinates is given. Find the corresponding rectangular coordinates. 

1. (2, −π ) 2. ⎛ 

−2, 3π ⎞ 

⎝ 

⎜ 

4 ⎠ 

⎟ 

Convert the rectangular equation to polar form. Assume a > 0. 

3. x 2 + y 2 = 4a 2 

4. y = 2a 

5. xy = −4a 

ANSWERS 

WU 1. about –1.37 2. about 1.56 3. x + y = 0 

4. x 2 + y 2 

4 = 1 5. (x − 2) 2 

9 

(y − 4)2 

+ = 1 

4 

QUIZ 1. 

(−2, 0) 2. ( 2, − 2) 3. r = 2a 

4. r = 2a 

sinθ 

4a 

5. r 2 = 

cosθ sinθ 

90 



SECTION 9.7 


A point in rectangular coordinates is given. Find the corresponding polar coordinates. 

1. (2, 0) 

2. (− 2, 2) 

Convert the polar equation to rectangular form. 

3. r = 5 

4. r = −2cscθ 

5. θ = 7π 6 


Test for symmetry with respect to θ = !/2, the polar axis, and the pole. 

1. r = 1+ 2cosθ 

2. r 2 = 4cos2θ 

Sketch the graph of the polar equation. 

3. r = 4 

4. r = 3sinθ 

2 

5. r 2 = 4cos2θ 



91

ANSWERS 

WU 1. (2, 0) 2. ⎛ 

2, 3π ⎝ 

⎜ 

4 

⎞ 

⎠ 

⎟ 3. x 2 + y 2 = 25 

QUIZ 

4. 

y = −2 5. y = 

3x 

3 

1. symmetry with respect to the polar axis 

2. symmetry with respect to the polar axis and θ = !/2 

3. 

4. 

5. 

92 



SECTION 9.8 


Sketch the graph of the polar equation. 

1. r = 4 − 3sinθ 2. r = 2sin2θ 

3. r = 

3 

2cosθ − sinθ 

4. r 2 = 9sinθ 


Identify the conic and sketch its graph. 

3 

1. r = 

1− sinθ 

2. r = 

4 

2 − 3sinθ 

5 

3. r = 

2 − cosθ 

Find a polar equation of a conic with the given eccentricity and directrix, and its focus 

at the pole. 

4. parabola: e = 1, x = –2 5. ellipse: e = 1 2 , y = –3 



93

ANSWERS 

WU 1. 2. 

3. 4. 

QUIZ 1. a parabola; 2. a hyperbola; 

3. an ellipse; 

2 

4. r = 

1− cosθ 

3 

5. r = 

2 − sinθ 

94

Warm-up and HW Quiz

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