Is the graph a function or a
relation?
Function
Function
Relation
State the domain of the
function:
x
y
1 x2
x 1
y 2
x 9
All real numbers All real numbers
except 1 or -1
except 3 or -3


x
y 2
x  5x
x
y
x 5
All real numbers
except 5
All real numbers
except 0 and 5
Find the composition
functions below:
f (x)  2x  5
f (x)  2x 2  x  2
g(x)  x 2
g(x)  x  3
( f og)(x) 
2x  5
2
2(x  3) 2  (x  3)  2
( f og)(x)  2(x 2  6x  9)  x  5

2x 2 12x 18  x  5
2x 2 11x 13

(g o f )(x) 
(2x  5) 2
(2x  5)(2x  5)
(g o f )(x)  (2x 2  x  2)  3

4 x 2  20x  25


2x 2  x  5
Find the x- and yintercepts:
x  2y 12  0
(12,0) and (0,6)

4 x  6y  24  0
(6,0) and (0,-4)
Find the zero of each
function:
f (x)  3x  2
f (x)  12x 2  48
0  12x  48
2
2
3

48  12x 2
4  x 2
x  4
x  2i
Dominic is opening a bank. He
determined that he will need $22,000
to buy a building and supplies to start.
He expects expenses for each following
month to be $12,300. Write an
equation that models the total expense
y after x months.
y  12,300x  22,000
Determine whether the graphs of the pair of
equations are parallel, coinciding, or
neither.
x - 2y = 12 and 4x + y = 20
3x - 2y = -6 and 6x - 4y = -12
1
y  x 6
2
y
Neither

y  4 x  20

3
x3
2
Coinciding

y
3
x3
2

Write an equation of the line that passes
through the points given:
(-2,4) and (6,-4)
m
(3,-5) and (0,4)
m
y 2  y1 8

 1
x 2  x1
8
y  y1  m(x  x1)

y  4  1(x  2)
y  4  x  2
y  x  2
y 2  y1 9

 3
x 2  x1 3
y  y1  m(x  x1)
y  5  3(x  3)
y  5  3x  9
y  3x  4
Write an equation of a line using the
information given.
1. No slope, (3,4)
2. slope = 3, (-3, -7)
y  y1  m(x  x1)
Slope is undefined
VERTICAL LINE
y  7  3(x  3)
y  7  3x  9
x3
y  3x  2

How can you tell if two
lines are perpendicular?

Their slopes are opposite reciprocals

HOW CAN WE TELL IF THEY ARE PARALLEL?

Their slopes are the SAME

Given f(x) and g(x),
find (f/g)(x)
f (x)  2x 2  3x
f (x)  4x 2  3x 10
g(x)  x  5
g(x)  6x 1
2x 2  3x
, x  5 
x 5

4 x 2  3x 10
1
,x 
6x 1
6
Solve this system of three
variables:
Find the product of each:
1 3 1 5 2

 

0
4
0
4
0

 

1 5 2 1 3

 

0
4
0
0
4

 

2X3
2X2
DOES NOT EXIST
1 7 2

 
0 16 0 

Evaluate the determinant of
this 3x3 matrix:
1 2 4
3 0 4
7 1 3
1 -2
3
0
-7 1
3 4 0
1
3 7
10 0 2
DOWNHILL - UPHILL
(0+56+12) - (0+4-18)
68 – (-14)
82

(18+280+0) - (0+0-8)
246+8
254
3
-4
1
3
-10 0
Evaluate each function
2
given: f (x)  2x  3x  2
1. f(a2)
2. f(3b4)
2(a 2 ) 2  3(a 2 )  2
2a
4
2(3b 4 ) 2  3(3b 4 )  2
 3a 2  2
18b 8  9b 4  2

Graph each function:
1. f(x) = 3x – 4
2. f(x) = -⅔x + 1
Find the values of x and y for
which the matrix equation is true.
x  y
x   1 3  y 
3x  2y
3x  2y  15
x  y 1
x  3 y
y  3x  6

I would use substitution:
(3  y)  y  1
3  2y  1
x  3  (1)
x 2
2y  2
y 1
y   15 3x  6
I would use substitution:

3x  2(3x  6)  15
3x  6x 12  15
y  3(3)  6
y  3
9x  27
(2,1)
x3
(3,3)
Given the two matrices, perform the
following operations.
A=
1.
1 6 1


0
3
2


3B

3 18 3


0
9
6


1 4 4 
B = 

11
0
50



2 0 1

2. 2A - C

Impossible

Find the inverse of each
matrix.
1.
1 3


4 7
1  7 3


19 4 1 
7
 19
4

 19
3 
19 
1  
19 
2.
2 3 


4 6
1 6 3


0 4 2 
Does not exist
Graph each inequality:
1. 2x + y – 3 < 0
2. x + 3y – 6 ≥ 0
Determine the intervals of increasing
and decreasing for each function:
f (x)  x 2  2x 1
f (x)  x 3  2x 2  x  4

Decreasing -1.5 < x < 0.2
Increasing x < -1.5, x > 0.2
Decreasing x < 1
Increasing x > 1
What lines are symmetric to
each function given:
1.
2.
x2 y2

1
4
9
(x  4) 2 (y  2) 2

1
4
9

x=4
y = -2
x=0
y=0
Graph each function and
it’s inverse.
1.
f (x)  x  3
2.
2
f (x)  x  2
f(x)

f(x)
f-1(x)
f-1(x)

Determine whether the critical pt given
is a max, min, or pt of inflection.
1. f (x)  3x 3 18x 2  4
(0.1,4.183)
(0,4)
(0.1,4.177)
MAX
x=0
2.
f (x)  3x 3  9x  5
(0.9,.913)
(1,1)
(1.1,.907)


MIN
x=1
Approximate the real zero.
1.
f (x)  x 3  2x 2  3x  5
x
y
-5 -65
-4 -25
-3 -5
-2
1
-1
-1
0 -5
1
-5
2
5
3
31
2. f (x)  x 4  8x 2 10
Rule of thumb: go
from -5 to 5 for your
x-values

So there is
zeroes
between 3 and -2, 2 and -1, 1
and 2
x
y
-5
-4
-3
-2
-1
0
1
2
3
435
138
19
-6
3
10
3
-6
19
If they want a decimal approximation, you need to make
another t-chart going by 0.1 in between these approximated
zeros.
So there is
zeroes
between 3 and -2, 2 and -1, 1
and 2
Or you could just
plug each answer and
see which one gets
you closest to a ZERO
Solve the system of
inequalities by graphing
x  2
y 0
x +y < 3
3x - y < 2
Use the related function to
find the min and max.
1.
f (x, y)  3x  2y
(2,3)(1,8)(0,5)

2.
l(x, y)  35x  20y 10
(3,3)(1,1)(0,2)
Determine the vertical
asymptotes of each function
x
f (x) 
5x
x 2
f (x) 
3x 1
VA: x = 0
VA: x = ⅓
2x  5
f (x)  2
x  4x
VA: x = 4, x = 0

Graph each rational
(x  2)(x  2)
x(x  5)
function
x 2
x2  4
f (x) 
x 2
x 2  5x
f (x) 
x
x


Hole at x = -2
Hole at x = 0

Find the roots of:
x  x 11x 10  0
3
A.)
3  i 29
2,
2
B.)

3  29
2,
2
USE THE
COMMON ROOT
AND DO

SYNTHETIC
DIVISION FIRST
2 IS COMMON
AMONG ALL THE
ANSWERS
C.)
2
2, -1
D.)
-2, 1
AFTER SYNTHETIC DIVISION,
TRY TO FACTOR, OR QUADRATIC FORMULA
TO FIND THE REST OF THE ROOTS.
Find the number of positive, negative,
and imaginary roots possible for this
function:
f (x)  2x 5  x 4  2x 3  x 10
f (x)  2x 5  x 4  2x 3  x 10
P
N
I
3
0
2
1
0
4
3, 1 positive roots
0 Negative roots
Each row adds up
to degree of
polynomial
In a polynomial equation, if there
is four changes in signs of the coefficients
of the terms, __________________________
there is 3 or 1 positive roots
Using Law of Sines
1. In ΔABC if A = 63.17°, b = 18, and a = 17, find B
2. In ΔABC if A = 29.17°, B = 62.3°, and c = 11.5, find a
Determine the type of discontinuity for
each function:
Find the maximum value for this
system of inequatilites:

Infeasible? Unbounded? Optimal solutions?
Solve this rational
inequality:

Use a number line
Find this trig value:
1. Given
Evaluate each problems
using the unit circle:
tan

4

2
tan

3
tan( 150) 

1
 3
3
3
Determine for each function
if it is odd, even, or neither?
Odd functions are
symmetric with respect
to the origin:
(a,b) and (-a,-b)
Even functions are
symmetric with respect
to the y-axis:
(a,b) and (-a,b)
y x
EVEN
x y 9
2
2
y  x3
y  x2
BOTH
ORIGIN
EVEN
List all possible rational
roots of each function:
x  2x  3x 10
3
2
P: 1, 2, 5, 10
Q: 1
4 x 3  x 2  5x  3
P: 1,3
Q: 1, 2, 4

1,2,5,10
1 3 1 3
1,3, , , ,
2 2 4 4
Use the triangles below to find missing
cos A, sin A, tan A
A
cos A 
8 89
89
sinA 
5 89
89
tanA 
5
8
89
8 ft.

5 ft.
Use the unit circle to find
each:
tan180 
0
sec 270 
undefined
5
sin

4
 2
2
csc(90)



-1
State the amplitude for each
function:

 
y  2sin 3  

4 
y  tan  45
Amplitude = none
Amplitude = 2

  
y  sec   3
3 2 
Amplitude = 1

Find the period for each
function:

 
y  2sin 3  

4 
y  tan  45
Period = π/k = π

Period = 2π/k
= 2π/3 or 120°
  
y  sec   3
3 2 

Period = 2π/k
= 6π or 1080°
Graph each function
1
f (x) 
x3
1
f (x) 
x 5
VA: x = 5
HA: y = 0
VA: x = -3
HA: y = 0
