Math 3
CHAPTER 4 & 7 TEST
Name:_______________________________________________
Evaluate the exponential expression.
1) - 642/3 = - 4 2 = - 16
Write the equation of the graph in its final position.
2) The graph of y = 8ex is translated 3 units to the left, reflected over the x-axis, and then translated 9 units
upward.
y = -8 e
(x+3)
+9
Find the inverse of the function.
3)
f(x) = e(- 2x) + 5
y = e -2x + 5
x = e -2y + 5
x - 5 = e -2y
ln x - 5 = - 2y
1
y = - ln x - 5
2
f-1 x = - ln
x-5
Find the domain of the function.
4) f(x) = -3log 8 (5x + 2)
5x + 2 > 0
5x > -2
2
x>5
domain = x| x > -
Evaluate the expression.
5) log (0.01) = log (10-2) = -2
Find the value of the logarithmic function.
6) log 8 (32)
32 = 8 x
25 = 23 x
25 = 2 3x
5 = 3x
5
=x
3
1
2
5
or -
2
,
5
Rewrite the expression as a sum or difference of logarithms or multiples of logarithms.
y 8
7) log 6
13x 3
log 6 y8
1
2
- log 6 13x 3
log 6 (y) + log 6 8
log 6 (y) +
1
2
- log 6 (13) + log 6 x 3
1
log6 (8) - log 6 (13) - 3log6 (x)
2
Solve the problem. Round answers to the nearest tenth.
8) If $4000 is invested in an account that pays interest compounded continuously, how long will it take to grow to
$5100 at 5%?
5100 = 4000e.05t
5100
= e.05t
4000
ln
51
= .05t
40
ln
51
40
=t
.05
t
4.9
9) Ben Franklin bequeathed $4000.00 to the city of Boston in 1790. Assuming the fund grew to $3 million in 200
years, find the interest rate that was compounded monthly that would yield this total value.
r 12(200)
3,000,000 = 4000 1 +
12
r (2400)
750 = 1 +
12
2400
2400
750 = 1 +
r
12
750 - 1 =
r
12
2400
r = 12
750 - 1
r 0.033
r 3.3%
Solve the equation. If necessary, round to the nearest thousandths.
10) e 3ln(x) = 17
3
e ln(x ) = 17
x3 = 17
3
x = 17
2.571
2
11) 3(10 - 2x) + 21
3(10-2x) = 81
= 102
3(10-2x) = 34
10-2x = 4
-2x = -6
x=3
12) log 2 12 = x
log 12
x=
3.585
log 2
1
= -3
13) log x
64
x-3 =
1
64
x3 = 64
3
x = 64
x=4
14) 5(x - 2) = 15
log 5 (x - 2) = log 15
(x - 2)log 5 = log 15
log 15
x - 2=
log 5
x=
x
log 15
+2
log 5
3.683
15) ln x = -7
-7
e =x
x
.0001
Solve the equation. Give an exact solution.
16) ln(4x - 5) = 2ln(5) - ln (x - 5)
ln(4x - 5) = ln(25) - ln (x - 5)
25
ln(4x - 5) = ln
x-5
4x - 5 =
25
x-5
(4x - 5)(x - 5) = 25
4x 2 - 20x - 5x + 25 = 25
4x 2 - 25x = 0
x(4x - 25) = 0
x=0
x=
25
4
25
4
3
17) log (x - 3) = 1 - log x
log(x - 3) + log(x) = 1
log[x(x - 3)] = 1
10 = x 2 - 3x
x2 - 3x - 10 = 0
(x - 5)(x + 2) = 0
x=5
x = -2
{5}
Use the base-change formula to find the logarithm to four decimal places.
log (39.03)
2.0451
18) log 6 (39.03) =
log (6)
Graph the function as accurately as possible.
1 (x - 5)
-8
19) f(x) =
3
4
20) f(x) = log
5
(x + 4) + 6
Write the equation in standard form. Find the line of symmetry, y-intercept, the directrix, and coordinates of the focus
and vertex. Determine whether the parabola opens upward, downward, to the left, or to the right. Identify each
concept. (10 points)
21) y = - 4x 2 + 24x - 41
y = -4 x2 - 6x + 9 - 41 + 36
y = -4 x - 3 2 - 5
l.o.s. x = 3
V(3, -5)
F(3, -5-
-4 =
1
4p
1
1
) = (3, - 5 )
16
16
Directrix y = - 5 +
-16p = 1
1
p=16
y-intercept (0, -41)
opens down
OR
h=-
24
=3
2(-4)
k = -4 3 2 + 24(3) - 41 = -4(9) + 72 - 41 = -36 + 31 = -5
Find the equation of the parabola determined by the given information.
22) Focus at (8, 2), directrix x = -4
8 - -4 12
1
1
-4 + 8 4
h=
p=
a=
= =2
=
=6
=
2
2
2
2
4(6) 24
Vertex is 2, 2
x=
1
y-22+2
24
5
1
16
1
15
=-4
+
=-4
16
16 16
16
Write an equation for the ellipse.
23) Foci (0,±2 7) and x-intercepts (±6, 0)
c = ±2 7
c2 = 4(7) = 28
b=±6
b2 = 36
28 = a 2 - 36
a2 = 64
x 2 y2
+
=1
36 64
6
Write the equation of the conic section in standard form.
Identify the conic.
Graph the conic.
If the conic section is a circle, also graph and name the center. Identify the radius.
If the conic section is a parabola, also graph and name the focus, vertex, and directrix.
If the conic section is an ellipse, also graph and name the foci and center.
If the conic section is a hyperbola, also graph and name the foci, center, and asymptotes.
24) x2 - y2 + 12x + 48y - 9 = 0
(10 points each)
x2 + 12x- y2 + 48y = 9
x2 + 12x+ 36 - (y2 - 48y + 576) = 9 + 36 - 576
x + 6 2 - y - 24 2 = - 531
y - 24 2
x+62
=1
531
531
c2 = 531 + 531
c = ± 1062 = ±3 118
hyperbola
center -6, 24
foci: - 6, 24 ±3 118
a=b=±
asymptotes: y - 24 = ± x + 6
531
not possible to graph
7
25) x2 + y2 - 12x + 18y + 117 = 81
x2 - 12x+ y2 + 18y = - 36
x2 - 12x + 36 + y2 + 18y + 81= - 36 + 36 + 81
x - 6 2 + y + 9 2 = 81
circle
8
center 6, -9
r=
81 = 9
26) 16x 2 + 9y2 + 128x - 90y + 337 = 0
16x 2 + 128x + 9y2 - 90y = - 337
16 x 2 + 8x + 16 + 9 y2 - 10y + 25 = - 337 + 256 + 225
16 x + 4 2 + 9 y - 5 2 = 144
x+42
y-52
+
=1
9
16
ellipse
center - 4, 5
foci: - 4, 5 ± 7
9
a = ± 16 = ± 4
b=± 9=±3
c = ± 16 - 9 = ± 7