CHAP 5 Review
1. 2x2 system
a. graph
b. algebra
2. 3x3 system
a. algebra
3. Solving Linear Inequalities by graph
Problems
1. Solve by graph, by elimination
2x + 3y = 8
3x + 4y = 13
2. Solve by algebra and matrix
x + 5y + x = -49
5x 3y + z = -9
14x 8y + 2z
MAT121 Chapter 1 Review
1. Simplify
a. Algebraic exponents
b. Polynomials
c. Complex numbers
2. Solving Equations
a. Linear
b. Quadratics
1.) Factoring
2.) Completing the Square
3.) Quadratic Formula
3. Solving Inequalities
4. Solving Story Problems
Book
MA121 TEST 4 REVIEW
1. Determining inverse functions
2. Proving inverse functions
3. Graphing exp functions
4. Write in log form
5. Write in exp form
6. Solve for x using logs and/or algebra
7. Simplify a log expression into one log
8. Simplify a log equa
MAT121 Test 2 Review
1. Midpoint
2. Distance between points
3. Slope and equation of a line
a. parallel slope
b. perpendicular slope
4. Equation of a circle
5. Graphing a line with intercepts
6. Identifying functions by graph
7. Identifying and sketching
MAT121 Test 3 Review
1.
2.
3.
4.
5.
6.
Graphing Quads
Evaluation using Synthetic division
Proving a value is a zero
Determining zeros starting with graphs
Sketching 3rd and 4th degree polynomials with end behavior
Determining polynomials with rational coe
Section 5.4 Linear Inequalities
3x + y
3x + y
3x + y
3x + y
>
<
4
4
4
4
y 2x + 6
15x + 3y < 12
2x y 5
x + 3y 1
x1
2x + y 7
x + 2y 5
x0
y0
P 546
#42
<, >
,
Solve by graph, test a point
Section 5.3 Solving Functional Systems
Identify the following:
x+y=4
x2 + y = 4
x2 + y2 = 4
Find the points of intersection
x2 + y2 = 25
-x + 2y = 5
Solve by graph and substitution
x2 - y = 8
x+y=4
Solve by elimination
x2 + y2 = 4
y + x2 = 5
Any way
x3 y
Section 5.1
Graphing:
Systems (2 variables)
4x 3y = 9
-2x + y = -5
2 different lines are said to be: INDEPENDENT
If they interest, they are:
CONSISTENT
Substitution:
4x + y = 4
y=x+2
Elimination:
2x 3y = 7
5x + 6y = 4
3x 2y = 8
-9y + 6y = 4
2 different li
Section 4.5
Exponential Applications
Annual Interest
A = P (1 + r)t
Compound Interest
A = P (1 + r/n)(nt)
Continual Interest
A = Pe(rt)
Growth Rate
Q(t) = Qoe(rt)
Decay Rate
Q(t) = Qoe(-rt)
Half Life
r = ln2/t
Problems:
You have $5500 to invest for 4 year
MA230 Section 5.2
2x 3y + 2z = 0
3x 4y + z = -20
x + 2y z = 16
3x 4y + z = -20
x + 2y z = 16
4x 2y
= -4
Add the 2 equations, z will drop
x + 2y z = 16 Multiply by 2 2x + 4y 2z = 32
2x 3y + 2z = 0
2x 3y + 2z = 0
4x + y
= 32
Add the 2 equations,
z will drop
Section 4.3 Logarithms
(Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations)
Exp form
Log form
y = bx
logby = x
25 = 5
log525 = 2
64 = 43
log39 = 2
logx125 = 3
Simplify:
log636 =
log327 =
log216 =
solve:
Section 3.8 Variations
Direct
y = kx
As x increases, y increases
As x decreases, y decreases
k
Inverse y = x
As x increases, y decreases
As x decreases, y increases
k is identified as the constant of variation
At the gas station: Total cost varies directl
Section 4.1 Functions
Relation: a set of ordered pairs
Function: a relation where each element of the domain is paired with exactly one
element in the range (for every x value, there is one and only one y value).
x
1
2
3
4
y
5
6
9
10
x
1
2
3
2
y
5
6
7
8
G
Section 3.5
Rational Functions
Vertical Asymptotes (a line a function will approach but never touch.)
The denominator of a rational function cannot equal zero
3x
f(x) = x + 4
x +2
x 4 x5
f(x) =
x and y intercepts (in a x intercept, y = 0
2
in a y intercep
Section 3.6
f(x) =
Polynomials with higher degree numerators
2
x 9
x3
Discontinuous at 3
2
x 9
x3
,
6,
D: x = 3
D: x is real number, x 3
f(x) =
Functions with a slant asymptote
f(x) =
x 2 +4
x
To determine the equation for the slant asymptote, divide the
Section 3.4 Graphing Polynomials
General forms: End Behavior
+
_
Sketch, describe end behavior, and identify the y intercept
y = - x3 + 4x 7
y = x4 + 3x2 5x + 6
Identify zeros (x intercepts)
y = x2 2x 8
Determine degree, describe end behavior, identify ze
Section 3.1 Quadratic Functions
y = x2
Forms:
y = a(x h)2 + k
1. vertex (h,k)
2. a determines: up/down
stretch/compress
y = ax2 + bx + c
1. y intercept (0,c)
2. x intercepts
x=
3. vertex (x value)
4. Axis of symmetry
b b24 ac
2a
b
x = 2a
,
b
(y value) f(
Section 3.3
3
4i
Polynomials with rational coefficients
-8
13
25
3 + 2i
If a polynomial has rational coefficients, and it has an irrational or a complex root
(zero) , the conjugate must also be a root.
If 7i is a root, -7i must be a root. If - 5 is a ro
Section 2.6
Basic Functional Graphs
NAME
Linear
Absolute
Value
Quadratic
Cubic
Square
Root
_
y = x
EQUATION
y=x
y = x
y=x
2
y=x
3
GRAPH
SPECIAL POINTS
none
vertex
vertex
inflection
(pivot)
initial
(node)
END BEHAVIOR
down,up
up,up
up,up
down,up
up
DOMAIN
Section 2.5 Analyzing Functional Graphs
Even functions
f(-x) = f(x)
Odd Functions
f(-x) = -f(x)
Examining a Function:
a.
b.
c.
d.
Intervals
Increasing, Decreasing, Constant
End behavior
(+) Functional value > 0,
(-) Functional value < 0
e. Zeros
f. Relati
Section 2.4
Functions
Function: A relation where one member of the domain (x) is paired with exactly on
member of the range (y).
Points: (4,2)(6,1)(7,8)
Charts:
Gallon
s
1
2
3
4
5
(5,1)(7,4)(5,8)
Cost
4
8
12
16
20
Ht (in)
58
60
64
72
72
(3,1)(7,1)(-2,1)
W
Section 2.2 Linear graphs/Slope of a line
1. Positive
2. Negative
3. Zero
4. No slope
Determine the slopes
Determine the intercepts
Determine slope from 2 points
m=
( y 2 y 1 )
( x 2x 1)
(x1, y1) (x2,y2)
(4,5) (6, -3)
(-1,3) (4,9)
Linear Equation (slope/i
CHAP 1 WORD PROBLEMS
1. At a cross country race, Katie jogged at 8 mph for the first
portion of the race and then ran at 12 mph for the remaining
portion. If the race was 21 miles and she finished in 2 hours,
how far did she jog?
2. My wife rode her bike
Section 2.3 Linear equations and graphs
y = mx + b
Determine the slope and the intercept
y = 5x 4
2x + 4y = 12
Determine the equation of the line given:
m=2
b = -8
m = -4 through the point (2,-5)
parallel to 2x + 3y = 8 through (6,5)
m= -
through the poin
Section 2.1 Relations
Set of ordered pairs (x,y)
Domain: set of all possible x values (inputs)
Range: set of all possible y values (outputs)
A = (5,6)(7,3)(7,8)
D(A) =
R(A)=
Teams = (Detroit, Lions)(Chicago, Bears)(Green Bay, Packers)
D(Teams) =
R(Teams)=