1)
Simplify each expression. Your final answer must contain positive
exponents only.
a)
4
16 x −20
81y −8
2
b) 15 y −5 (5 x −3 y ) −1
2
2)
Rationalize the numerator and simplify.
3+ x
x +1
3)
2
Factor the following polynomials as much as possible.
a)
x 2 y − 4 x + 3xy 2 − 12 y
b)
27 x 3 + 125 y 3
c)
12 x 2 + 7 x − 10
2
2
2
4)
Perform the indicated operations and simplify.
2x 6
−
x
3
a)
x+3
3
x 2 − 36 x 2 + 12 x + 36
÷
b)
4 x − 12
x−3
3
5)
Use long division to find the quotient and remainder of
x − 6x 2 + 5x + 7
.
x −1
3
3
6a)
Find the equation of a line parallel to 7 x − 14 y = 27 and passing
through the point (8, 14).
b)
c)
What are the x and y intercepts of the line y = −3 x + 15 ?
2
1
Calculate the distance between the points (1, -5) and (-4, 10).
1
d)
Calculate the midpoint of (1, -5) and (-4, 10).
1
e)
Find the equation of a horizontal line passing through the point (-4,10)
1
7) Solve each of the following for x.
a ) 2 − (3x + 4) = −3(5 x − 2) + 2 x
3
7 cont.) Solve each of the following for x.
b) x 4 − 3 x 3 − 10 x 2 = 0
3
c ) 4 x 2 − 12 x = −1
3
d)
5
3
4x +1
−
= 2
x+2 x−3 x − x−6
3
7 cont.) Solve each of the following for x.
3
e) 2 + 3 x + 1 = 7
f ) 3( 4 − x ) − 1 < 6( x + 2)
3
8) Given f ( x ) = 3 x − x
f ( x + h) − f ( x )
find and simplify
h
2
9) Given f ( x) = x + 2 , find
a) the domain of f (x)
1
1
b) the range of f (x)
1
9 cont) Given f ( x) =
c) f −1 ( x )
d) Using g ( x) =
x + 2 , find
x
find
x−4
1
( g o f )( x) = g ( f ( x))
10) Given the quadratic function
a) find the vertex
f ( x) = x 2 − 2 x − 8
b) find the equation of the axis of symmetry
c) find the y-intercept
d) find the x-intercept(s)
e) sketch the graph of f (x) and
label the information found above
f) state the range of f (x)
1
4
− x 2 + 2
f
x
(
)
=
11) Given

3 x − 1
x<2
x≥2
4
a) find f (4)
b) find f (1)
c) sketch the graph of f (x)
12) Given the rational function f ( x) =
2x + 6
x −1
a) find the x-intercept
b) the y-intercept
c) the equation of the vertical asymptote
d) the equation of the horizontal asymptote
e) sketch the graph of f (x) and
label the information found above
f) state the domain of f (x)
4
13) Calculate to 4 decimal places:
log3 18
14) Sketch the graph of
Indicate the
Domain
1
y = 2x – 16
3
Range
x-intercept
y-intercept
horizontal asymptote
15) Derek knows what he likes and he knows that what he likes is not
cheap. So he is saving up… He put the $7500 he made last week into
a GIC that pays interest at 5% compounded semi-annually for 6 years.
What will it be worth then?
3
16) Write as sums(s) and difference(s) of simple logarithms:
ln
20 y
x
2
17) Write using one logarithm
ln 13 + 4 ln x – ln(x – 3)
18) Solve for x:
5(2x ) = 6000
19) Find x:
log(x + 6) – log(x – 2) = 0.69897
2
3
3
20) Calculate to 4 decimals
sec 7°
1
cos-1 ( 0.5 )
1
 3π 

 4 
21) The exact value of tan 
22) Write
5π
radians in degrees.
3
1
1
23) A is an angle in Quadrant III with tan A = 1.5 What is angle A ?
1
24) Find 2 values of B (between 0 and 360°) with sin B = 0.5
1
25) Prove :
sin D
4
+ 3 tan D =
cos D
cot D
2
26) Prove :
1
1
−
= 2 sec E tan E
1 − sin E 1 + sin E
2
27) Find the size of angle X:
3
28) Find x :
3
29) Jonathan has a ladder that is 7 m long. He leans it against the side of
the house, so the top of the ladder is on a window sill 5 m above ground. The
base of the ladder is 4 m from the side of the house. Trouble is: the ground
is not perpendicular to the wall of the house. What angle does the ground
make with the side of the house ?
3
30) Sketch the graph of y = 6 cos 3x
What is the Amplitude ?
What is the Period ?
2
Answers
2 y2
3x 5
3x3
y6
9− x
( x + 1) (3 − x )
(3x + 5y) ( 9x2 – 15xy + 25y2)
(4x + 5) (3x – 2)
2( x − 3)
3x
x−6
4( x + 6)
x2 – 5x rem 7
y = ½ x + 10
x-intercept: (5, 0)
y-intercept: (0, 15)
15.8113
( -1.5 , 2.5 )
y = 10
x = 0.8
x = 0, 5 or -2
x = 2.9142 or 0.0858
x = - 11
x=8
x > - 1/9
6x + 3h – 1
Domain : x ≥ 0
Range : y ≥ 2
y = (x – 2)2
x +2
x −2
Vertex : (1, -9)
Axis of symmetry: x = 1
y-intercept: y = -8
x-intercepts: 4, -2
Range y ≥ -9
f(4) = 11
f(1) = 1
x-intercept : x = -3
y-intercept : y = -6
vertical asymptote x = 1
horizontal asymptote y = 2
domain : x ≠ 1
2.6309
Domain : all real numbers
x-intercept: 4
y-intercept: -15
horizontal asymptote y = -16
$10 086.67
ln 20 + ln y - ½ ln x
 13 x 4 

ln
x
−
3


x =10.2288
x=4
1.0075
60°
-1
300°
236.31°
30° or 150°
proof using tan = sin/cos
and
proof using common denominator,
and 1 – sin2 = cos2
30.95°
3.5542
101.54°
Amplitude = 6
Period = 120°
cot = cos / sin
sec = 1/cos
tan = sin/cos