Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Match the equation to the graph.
1)
y2
4–x2
9= 1
1)
A)
B)
C)
D)
2)
y2= – 9x
2)
A)
B)
C)
D)
3)
x2=9y
3)
A)
B)
C)
D)
4)
x2= – 10y
4)
A)
B)
C)
D)
5)
(y + 2)2=5(x + 2)
5)
A)
B)
C)
D)
5
6)
y2=9x
6)
A)
B)
C)
D)
7)
x2
4–y2
16 = 1
7)
A)
B)
C)
D)
8)
(x – 2)2=7(y – 2)
8)
A)
B)
C)
D)
Convert the equation to the standard form for a parabola by completing the square on x or y as appropriate.
9)
y2– 4y + 4x + 0 = 0
9)
A)
(y – 2)2= – 4(x – 1)
B)
(y – 2)2= – 4(x + 1)
C)
(y + 2)2=4(x – 1)
D)
(y + 2)2= – 4(x – 1)
8
Find the standard form of the equation of the hyperbola satisfying the given conditions.
10)
Endpoints of transverse axis: (0, –8), (0, 8); asymptote: y =4
3x
10)
A)
y2
9–x2
16 = 1
B)
y2
36 –x2
64 = 1
C)
y2
64 –x2
9= 1
D)
y2
64 –x2
36 = 1
Determine the direction in which the parabola opens, and the vertex.
11)
x = – (y + 2)2– 4
11)
A)
Opens to right; (–2, –4)
B)
Opens to left; (–4, –2)
C)
Opens to left; (–2, –4)
D)
Opens to left; (–4, 2)
Find the focus and directrix of the parabola with the given equation.
12)
x2= – 36y
12)
A)
focus: (0, 9)
directrix: y = – 9
B)
focus: (0, –9)
directrix: y = – 9
C)
focus: (–18, 0)
directrix: x =9
D)
focus: (0, –9)
directrix: y =9
13)
x2=4y
13)
A)
focus: (1, 0)
directrix: x =1
B)
focus: (0, –1)
directrix: x = – 1
C)
focus: (0, 1)
directrix: y = – 1
D)
focus: (1, 0)
directrix: y =1
Graph the ellipse and locate the foci.
9
14)
x2
49 +y2
25 = 1
14)
A)
foci at ( 39, 0) and (–39, 0)
B)
foci at (0, 2 6) and (0, –2 6)
C)
foci at (0, 39) and (0, –39)
D)
foci at (2 6, 0) and (–2 6, 0)
Find the location of the center, vertices, and foci for the hyperbola described by the equation.
15)
(x – 4)2–100(y – 1)2=100
15)
A)
Center: (–4, –1); Vertices: (–14, –1) and (6, –1); Foci: (–4–101, 1) and (–4+101, 1)
B)
Center: (4, 1); Vertices: (–5, 2) and (15, 2); Foci: (5–101, 2) and (5+101, 2)
C)
Center: (4, 1); Vertices: (–6, 1) and (14, 1); Foci: (4 –101, 1) and (4+101, 1)
D)
Center: (4, 1); Vertices: (10, 1) and (–10, 1); Foci: (–101, 1) and ( 101, 1)
Find the standard form of the equation of the parabola using the information given.
16)
Focus: (–3, 0); Directrix: y = – 4
16)
A)
(y + 2)2=8(x + 3)
B)
(x + 3)2=8(y + 2)
C)
(x + 2)2=8(y + 3)
D)
(y + 3)2=8(x + 2)
Find the foci of the ellipse whose equation is given.
17)
25(x + 1)2+36(y + 3)2=900
17)
A)
foci at (–3+11, –1) and (–3–11, –1)
B)
foci at (–11, –3) and ( 11, –3)
C)
foci at (–1+11, –1) and (–1–11, –1)
D)
foci at (–1+11, –3) and (–1–11, –3)
Find the focus and directrix of the parabola with the given equation.
18)
y2= – 16x
18)
A)
focus: (4, 0)
directrix: x = – 4
B)
focus: (–4, 0)
directrix: y =4
C)
focus: (–4, 0)
directrix: x =4
D)
focus: (0, –4)
directrix: y =4
Find the standard form of the equation of the ellipse and give the location of its foci.
19)
19)
A)
x2
4+y2
16 = 1
foci at (0, 4) and (2, 0)
B)
x2
4+y2
16 = 1
foci at (0, –4) and (0, 4)
C)
x2
4+y2
16 = 1
foci at (0, –2 3) and (0, 2 3)
D)
x2
16 +y2
4= 1
foci at (0, –2 3) and (0, 2 3)
Find the location of the center, vertices, and foci for the hyperbola described by the equation.
20)
(y + 4)2
9–(x + 2)2
36 = 1
20)
A)
Center: (–2, –4); Vertices: (–2, –7) and (–2, –1); Foci: (–2, –4–3 5) and (–2, –4+3 5)
B)
Center: (–2, –4); Vertices: (–2, –4–3 5) and (–2, –4+3 5); Foci: (–2, –7) and (–2, –1)
C)
Center: (–2, –4); Vertices: (7, –6) and (–1, 0); Foci: (7, –3–3 5) and (–1, –3+3 5)
D)
Center: (2, 4); Vertices: (2, 1) and (2, 7); Foci: (2, 4–3 5) and (2, 4+3 5)
Use the relation’s graph to determine its domain and range.
21)
y2
9–x2
4= 1
21)
A)
Domain: (–, –3] or [3, )
Range: (–, )
B)
Domain: (–, –3] and [3, )
Range: (–, )
C)
Domain: (–, )
Range: (–, –3] or [3, )
D)
Domain: (–, )
Range: (–, –3] and [3, )
Graph the ellipse.
22)
(x + 1)2
16 +(y – 2)2
9= 1
22)
13
A)
B)
C)
D)
Solve the problem.
23)
The arch beneath a bridge is semi–elliptical, a one–way roadway passes under the arch. The width
of the roadway is 36 feet and the height of the arch over the center of the roadway is 13 feet. Two
trucks plan to use this road. They are both 12 feet wide. Truck 1 has an overall height of 12 feet and
Truck 2 has an overall height of 11 feet. Draw a rough sketch of the situation and determine which
of the trucks can pass under the bridge.
23)
A)
Truck 2 can pass under the bridge, but Truck 1 cannot.
B)
Truck 1 can pass under the bridge, but Truck 2 cannot.
C)
Neither Truck 1 nor Truck 2 can pass under the bridge.
D)
Both Truck 1 and Truck 2 can pass under the bridge.
Use the center, vertices, and asymptotes to graph the hyperbola.
14
24)
(y – 2)2–4(x – 3)2=4
24)
A)
B)
C)
D)
Solve the problem.
25)
An experimental model for a suspension bridge is built. In one section, cable runs from the top of
one tower down to the roadway, just touching it there, and up again to the top of a second tower.
The towers are both 16 inches tall and stand 80 inches apart. Find the vertical distance from the
roadway to the cable at a point on the road 16 inches from the lowest point of the cable.
25)
A)
2.76 in.
B)
2.56 in.
C)
10.24 in.
D)
2.36 in.
Find the vertices and locate the foci for the hyperbola whose equation is given.
26)
y = ± x2–7
26)
A)
vertices: (–7, 0), (7, 0)
foci: (–14, 0), ( 14, 0)
B)
vertices: (–7, 0), ( 7, 0)
foci: (–14, 0), ( 14, 0)
C)
vertices: (–7, 0), (7, 0)
foci: (–7, 0), ( 7, 0)
D)
vertices: (0, –7), (0, 7)
foci: (0, –14), (0, 14)
Find the foci of the ellipse whose equation is given.
27)
(x – 1)2
36 +(y + 3)2
25 = 1
27)
A)
foci at (1+11, –3) and (1–11, –3)
B)
foci at (–3+11, 1) and (–3–11, 1)
C)
foci at (1+11, 1) and (1–11, 1)
D)
foci at (–11, –3) and ( 11, –3)
Find the standard form of the equation of the ellipse satisfying the given conditions.
28)
Major axis vertical with length 14; length of minor axis =12; center (0, 0)
28)
A)
x2
49 +y2
36 = 1
B)
x2
36 +y2
49 = 1
C)
x2
12 +y2
49 = 1
D)
x2
144 +y2
196 = 1
Find the vertices and locate the foci for the hyperbola whose equation is given.
29)
81y2–4x2=324
29)
A)
vertices: (–2, 0), (2, 0)
foci: (–85, 0), ( 85, 0)
B)
vertices: (0, –9), (0, 9)
foci: (0, –85), (0, 85)
C)
vertices: (0, –2), (0, 2)
foci: (0, –85), (0, 85)
D)
vertices: (–9, 0), (9, 0)
foci: (–77, 0), ( 77, 0)
Convert the equation to the standard form for an ellipse by completing the square on x and y.
30)
4x2+36y2– 16x + 72y – 92 = 0
30)
A)
(x + 2)2
36 +(y – 1)2
4= 1
B)
(x – 2)2
36 +(y + 1)2
4= 1
C)
(x – 2)2
4+(y + 1)2
36 = 1
D)
(x + 1)2
36 +(y – 2)2
4= 1
Graph the parabola with the given equation.
31)
(x + 2)2= – 5(y + 1)
31)
A)
B)
17
C)
D)
Find the standard form of the equation of the ellipse satisfying the given conditions.
32)
Endpoints of major axis: (–2, –10) and (–2, 2); endpoints of minor axis: (–7, –4) and (3, –4);
32)
A)
(x + 2)2
25 +(y + 4)2
36 = 1
B)
(x + 4)2
25 +(y + 2)2
36 = 1
C)
(x – 2)2
25 +(y – 4)2
36 = 1
D)
(x – 5)2
25 +(y – 6)2
36 = 1
Find the standard form of the equation of the parabola using the information given.
33)
Focus: (0, –23); Directrix: y =23
33)
A)
y2= – 23x
B)
y2= – 92x
C)
x2= – 92y
D)
x2=92y
Graph the ellipse and locate the foci.
18
34)
x2
9+y2
25 = 1
34)
A)
foci at (3 3, 0) and (–3 3, 0)
B)
foci at (4, 0) and (–4, 0)
C)
foci at (0, 4) and (0, –4)
D)
foci at (0, 3 3) and (0, –3 3)
Find the standard form of the equation of the parabola using the information given.
35)
Focus: (2, 0); Directrix: x = – 2
35)
A)
x2=8y
B)
y2= – 8x
C)
y2=2x
D)
y2=8x
Use the vertex and the direction in which the parabola opens to determine the relation’s domain and range.
36)
y2– 4y – x + 3 = 0
36)
A)
Domain: (–, )
Range: (–, 1]
B)
Domain: (–, 1)
Range: (–, )
C)
Domain: (–1, ]
Range: (–, )
D)
Domain: (–, )
Range: (–, )
Find the standard form of the equation of the hyperbola satisfying the given conditions.
37)
Center: (6, 2); Focus: (3, 2); Vertex: (5, 2)
37)
A)
(x –2)2
8–(y –6)2= 1
B)
(x –6)2–(y –2)2
8= 1
C)
(x –6)2
8–(y –2)2= 1
D)
(x –2)2–(y –6)2
8= 1
Find the standard form of the equation of the ellipse satisfying the given conditions.
38)
Major axis horizontal with length 20; length of minor axis =10; center (0, 0)
38)
A)
x2
25 +y2
100 = 1
B)
x2
20 +y2
25 = 1
C)
x2
400 +y2
100 = 1
D)
x2
100 +y2
25 = 1
Use vertices and asymptotes to graph the hyperbola. Find the equations of the asymptotes.
20
D)