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Math 102
Exam 3: Study Guide
Section 2.5
Section 3.4
Write the equation of each circle. Express the …nal
For problems 16 & 17:
equation in the form x2 + y 2 + Dx + Ey + F = 0
1)
Center at ( 3; 4) and r = 2
2)
Center at ( 1; 5) and r = 3
b
2a ;
(i) Find the vertex by using
f
b
2a
.
(ii) Find the axis of symmetry and x intercept.
(iii) Graph the function.
Find the center and the radius of each circle and then
graph it.
x2 + y 2
3)
2
6x
10y + 30 = 0
3x2 + 6x
16)
f (x) =
17)
f (x) = x2 + 3x
5
1
2
4)
x + y + 10x + 14y + 73 = 0
5)
x2 + y 2
10x = 0
Find the equation of the circle with the given condi-
Applications.
18)
tions:
6)
The circle passes through the origin and has its center
7)
The line segment determined by the points (3; 4) and
at (6; 8) :
of their squares is a minimum
19)
Find two numbers whose sum is 50 and whose product
is a maximum.
( 3; 2) is a diameter.
Section 3.3
Section 8.1
Find the vertex, x intercept, y intercept, and axis of
symmetry. Then graph the quadratic function.
f (x) =
9)
f (x) = (x + 2)
parabola, and sketch the graph.
2
For problems 10-12:
(i) Change the form f (x) = ax2 + bx + c to the form
2
h) + k:
(ii) Find the vertex (h; k) and axis of symmetry.
For problems 20-23, use the formulas from section 8.1
to …nd the vertex, focus, and directrix of the given
x2 + 2
8)
f (x) = a (x
Find two numbers whose sum is 20 such that the sum
20)
y 2 = 8x
21)
x2 =
22)
x2 + 6x + 8y + 1 = 0
23)
y 2 + 6y
12y
4x + 1 = 0
(iii) Graph the function.
10)
f (x) = 2x2
x
2
2x
11)
f (x) =
12)
f (x) = 2x2 + 12x + 17
2x + 1
Graph each piecewise-de…ned
function.
(
x2
for x < 0
13) f (x) =
x2 + 3 for x 0
8
2
>
< x for x < 1
14) f (x) =
x
for
1 x<1
>
: 2
x
for x 1
8
>
for x < 0
< 2x
2
15) f (x) =
x
for 0 x < 2
>
: 1
2
2 x + 2 for x
Find an equation of the parabola that satis…es the
given conditions.
24)
Focus (0; 3), directrix y =
25)
Focus ( 1; 0), directrix x = 1
26)
Focus (3; 4), directrix y =
27)
Focus ( 4; 5), directrix x = 0
Page: 1
3
2
Bibiana Lopez
Math 102
Exam 3: Study Guide
Section 8.4
Section 8.2
Find the vertices, the endpoints of the minor axis, For problems 45-48:
and the foci of the given ellipse, and sketch its graph. (i) Graph the system to …nd an approximate
28)
solution.
x2 + 4y 2 = 4
(ii) Solve the system by the substitution or
29) 16x2 + 4y 2 = 64
30)
31)
4x
2
8x + 9y
2
2
36y + 4 = 0
45)
2
4x + 16x + y + 2y + 1 = 0
46)
Find an equation of the ellipse that satis…es the given
conditions.
32)
Vertices ( 4; 0) ; Foci ( 2; 0)
33)
Vertices (0; 3) ; Foci (0; 2)
47)
48)
elimination method.
!
x2 + y 2 = 5
x + 2y = 5
y = x2 + 6x + 7
2x + y =
2x
5
y=7
!
!
3x2 + y 2 = 21
!
x y=2
x2
y 2 = 16
Section 8.3
Find the vertices, the foci and the equation of the
asymptotes, and sketch each hyperbola.
34)
4x2
9y 2 = 36
35)
9y 2
16x2 = 144
36)
4x2
24x
37)
y2
4y
9y 2
4x2
18y
24x
9=0
36 = 0
Find an equation of the hyperbola that satis…es the
given conditions.
38)
Vertices ( 2; 0) ; Foci ( 3; 0)
39)
Vertices (0; 3) ; Foci (0; 5)
Identify the graph of each of the equations as a
straight line, a circle, a parabola, an ellipse, or a hyperbola. [Do not sketch the graphs].
40)
x2
7x + y 2 + 8y
2
41)
5x + 3x
42)
5x
43)
4x
44)
x2
2y
2
2=0
3y
1=0
7y = 9
2
x + y 2 + 2y
3x + y
3=0
4=0
Page: 2
Bibiana Lopez