Jim Jack (J²)
MATH 1314 - College Algebra
Admin
Class
Grading
Attendance
1.1 Numbers, Data, and Problem Solving
Natural Numbers
Integers
Rational
Irrational
PEMDAS
1. Parenthesis
2. Exponents
3. Multiplication and Division, left to right
4. Addition and Subtraction, left to right
3(1  5)  4
2
2
10  6
4 72
53
Scientific Notation
A real number r is in scientific notation when r
is written as c 10n , where 1  c  10 and n is an
integer.
Dist to sun = 93,000,000
Pop of world in 2050 = 9,000,000,000
1
Light travels in 1 mile =
186,000
Mass of the earth – 5.98 1024 kg
3 10 2 10 
3
4
5 10 6 10 
3
4.6 10 1
2 10 2
5
 6 103 
2


1
.
2

10


6
 4 10 
103  450 

4500 

 0.233 
Calculator Exercise
3
131
 3  1.22
1 2
3.7  9.8
36
3
Find the speed the earth travels in space relative
to the sun in miles per hour.
C  2r
Find the volume of a soda can with radius 1.4”
and height 5”.
Can it hold 16oz?
V   r 2h
1"3  0.55oz
1.2 Visualizing and Graphing Data
Low temperatures in Minneapolis for six nights
-12
-4
-8
21
18
9
maximum, minimum, mean, median
A relation is a set of ordered pairs.
Portland rainfall
1 2 3 4 5 6 7 8 9 10 11 12
6.2 3.9 3.6 2.3 2.0 1.5 0.5 1.1 1.6 3.1 3.2 6.4
Domain=
20
1.2
Domain=
Range=
20
1.1
40
1.5
Range=
40
1.6
Portland rainfall
1 2 3 4 5 6 7 8 9 10 11 12
6.2 3.9 3.6 2.3 2.0 1.5 0.5 1.1 1.6 3.1 3.2 6.4
Portland Rainfall
7
6.4
6.2
6
Inches
5
4
3.9
3.6
3.1 3.2
3
2.3
2
2
1.6
1.5
1.1
1
0.5
0
0
1
2
3
4
5
6
Months
7
8
9
10
11
12
Distance formula
 1,3 4,3
3,4  2,7 
Distance between two moving cars
100
50
KC
50
100
Linear Approximation
Midpoint formula
 3,5
 3,1  1,3
The midpoint of a line segment with endpoints
 x1 , y1  and  x2 , y2  in the xy plane is
 x1  x2 , y1  y2 


2 
 2
6,7   4,6
Population of the US
In 1990, the population of the US was 249
million, and in 2010, it was 308 million.
Find the population in 2000.
A circle is the set of points equidistant from a
center point.
r 2  x2  y 2
r 2  x  h   y  k 
2
Graph x 2  y 2  9
 x  12   y  22  4
2
Finding the equation of a circle
a) Radius = 4, Center is  3,5
b) center 6,3 with a point on the circle 1,2 
A line is drawn from the point  3,4 to the
point 5,6 . State the equation of a circle for
which this line is the circle’s diameter.
Completing the square
2
k 
k

2
x  kx      x  
2
2 
skip
2
x 2  8x
Find the center and the radius of the circle:
x 2  y 2  6 x  4 y  12  0
1.3 Functions and their representations
Thunder and lightning speed of sound=1050 ft/sec
5
1
10
2
15
3
f x  y
20
4
x
f
Rule: Compute y by dividing x by 5
f x  x
5
meaningful input – domain
corresponding output – range
25
5
f(x)
Verbal: Divide x by 5 to obtain y miles
Numerical:
1
2
0.2
0.4
3
0.6
4
0.8
5
1.0
6
1.2
7
1.4
Diagrammatic:
5
1
a
w
A
x
10
2
b
x
B
y
15
3
c
y
C
z
20
4
z
D
symbolic: f  x   x
f 6 
graphical:
5
Leonhard Euler
(1707-1783)
Graph f  x   x
A function is a relation in which each element in
the domain corresponds to exactly one element
in the range.
 x  exactly one y  function (rule for finding y)
1
 x
x
Ordered pairs
f  1  4 f 0  3
Domain
Range
f 1  4
f 2  2
x
Let f  x  
x 1
f 2 
f 1 
f a 1 
Domain
g x  x2  2x
Domain
Range
Find domain and range graphically
f x  x  2
Air cools at a rate of 3.6º F per 1,000 ft altitude
Figure for 0  x  6 , where x is in thousands ft.
Verbal:
Symbolic:
Graphical:
Numerical:
Crutch length – trial and error – or f  x   0.72 x  2
(65” height) skip ex
Symbolic:
Graphical:
Numerical:
Identifying functions
A   2,3,  1,2, 0,3,  2,4
B  1,4, 2,5,  3,4,  1,7 , 0,4 
Vertical line test
1.4 Types of Functions and Their Rates of Change
Describe data
Linear
make predictions
Non-linear
Constant
Windspeed at Hilo, HI
May June July Aug Sept Oct Nov Dec
7
7
7
7
7
7
7
7
f=
A function represented by f  x   b , where b is a
fixed number, is a constant function.
Discrete function
Continuous function
Other constant fcns – Thermostat, speed control
Linear functions
A car is initially located 30 miles N of the Red
River on I-35 traveling northbound at 60mph.
0
30
1
90
2
150
3
210
4
270
5
330
f=
A function represented by f  x   ax  b , where
a and b are fixed, is a linear function.
f  x   1.5x  6
f x   8x
f  x   72
f  x   1.9  3x
Wages earned at $8.25/hr
Tuition at $75/credit hr + $55 student fee
Distance traveled by light at 186,000 miles/sec.
Slope as a rate of change
The slope of the line passing through the points
 x1 , y1  and  x2 , y2  is
y y2  y1
m

 x1  x2 
x x2  x1
Positive slope
Negative slope
No slope
Find the slope of a line through  2,3 and 1,2 
Songs that can be stored on x GB of Ipod memory
20,000
songs
80Gb
Pass thru origin?
Slope?
Rate of change?
Non-linear functions
x
f(x)
0
0
1
1
2
4
average temperature in each month
height of a child at age 2 through 18
3
9
4
16
Linear and non-linear data
0
-4
5
-2
10
0
15
2
20
4
-3
5
0
7
3
10
6
14
9
19
0
11
1
11
2
11
3
11
4
11
0
3
1
6
3
9
6
12
10
15
Recognizing linear functions
f x  6  4x
f  x   3x 2  2
f x  5
skip
Increasing and Decreasing Functions
f x  x
f  x   x3
f x  x
Suppose f is a function defined over interval I.
For x1 , x2  I :
a) f increases on I if x1  x2  f  x1   f  x2 
b) f decreases on I if x1  x2  f  x1   f  x2 
Interval Notation
2 x 2
x 2
Average rate of change, non-linear function
m
y2  y1 f  x2   f  x1 

x2  x1
x2  x1
f  x   2x 2
rate of change from x = 1 to x = 3
rate of change of US pop, 1800 – 1840, 1900 – 1940
Year Population
1800
5
1840
17
1900
76
1940
132
The difference quotient
m
f x  h  f x  f x  h  f x

x  h  x
h
Find f  x  h  first, then difference quotient.
f  x   3x  2
f x  x2  2x
Ch 2 Linear Functions and Equations
2.1 Equations of lines
Point-slope form
y  y1
m
x  x1
x intercept  4,0 , y intercept 0,2 
Slope  1 passing through  3,7 
2
Equation of a line passing through  2,3, 1,3
Alternate method slope-intercept form
Equation of a line passing through  2,1, 2,3
y  mx  b
Apple sold 55m iPods in 2008, and 43m in 2011
f x 
f 2010 
f 2023 
Determine x and y intercepts
ax  by  c
3x  4 y  12
4x  3y  6
Equations of horizontal and vertical lines
An equation of a horizontal line with y intercept
is y  b . An equation of the vertical line with x
intercept is x  k .
Vertical and horizontal lines through 8,5
Finding parallel lines
Two lines with slopes m1 and m2 , neither of
which are vertical, are parallel if and only if
m1  m2 .
Find the equation of a line parallel to the line
y  2 x  5, passing through 4,3.
Finding perpendicular lines
Two lines with slopes m1 and m2 , neither of
which are vertical, are perpendicular if and only
if their slopes have product  1, i.e. m1m2  1.
Find the equation of a line perpendicular to the
line y   2 x  2 , passing through  2,1.
3
Determining a rectangle
Interpolation and Extrapolation
In 2005, about $350M was spent on US musical
downloads. This amount reached $1350M in
2010. Model, find value in 2008, 2003.
2003
2005
2008
2010
350
1350
Investments in cloud computing
Year
2005 2006 2007 2008 2009
Investments 26 113 195 299 374
Model the data
Estimate Investments in 2014
2.2 Linear Equations
Equation – statement that two expressions are equal
Solve – find variable value(s) that make stmt true
Solution set – set of the values that make it true
x  x2 1
x2 1  0
Equivalent – two stmts with same solution set
Contradiction – equation with no solution set
x2 x
Identity – equation with infinite solution set
x  x  2x
A linear equation in one variable is an equation
that can be written in the form
ax  b  0, a, b  , a  0
(an equation that is not linear is called non-linear)
Addition property
a b  ac bc
Multiplication property
a  b  ac  bc
Solving linear equations symbolically
3 x  4  2 x  1
Solving linear equations symbolically
32 x  5  10   x  5
Eliminating fractions and decimals
x
2
1 
3
3
t 2 1
1
 t  5  3  t 
4
3
12
5.1x  2  3.7
0.03 z  3  0.52 z  1  0.23
The linear function f  x   4 x  2008  55
estimates iPod sales (million) during year x.
When will sales reach 27 million?
Contradictions, Identities, Conditional Equations
7  6 x  23 x  1
2 x  5  3  1  2 x 
25  x   25  3 x  5  5 x
An equation with one expression on each side of
the = sign can be solved by the intersection of
graphs method.
Using a graphing calculator;
1. Set Y1 equal to the left side, and Y2 equal to
the right side of the equation.
2. Graph both on one set of coordinates
3. Points of intersection are ordered pairs that
make both sides true at the same time.
These ordered pairs solve the equation.
2x 1  1 x  2
2
The market share of music on CDs held from
1987 to 1998 could be modeled by
f  x   5.91x  13.7 . During the same period,
cassette tape sales could be modeled by
f  x   4.71x  64.7 . (x=0 indicates year 1987)
32 x     13 x  0 Y1 
32 X     X / 3
Percentages: f  x   P x 
Suppose 76% of all bicycle riders do not wear
helmets. There are 38.7 million riders that do
not wear helmets, Find total ridership.
Solving for a variable
C  2 r
1
A  h a  b 
2
Problem Solving
1. Read problem, understand it. Assign
variable to what you must find. Write
other quantities in terms of this variable.
2. Model it, write the equation. Diagram it if
necessary.
3. Solve the equation, state solution
4. Check your work.
Work problems
A pump can empty a gasoline tank in 5 hours.
A smaller pump can empty the same tank in 9
hours. How long will it take both pumps
working together to empty the tank?
Motion problems
In one hour, an athlete runs 10.1 miles by
running some at 8mph, the rest at 11 mph. How
long did the athlete run at each speed?
Similar triangles
A person 6 feet tall stands 17 feet from a
streetlight, and casts an 8 ft shadow. Estimate
the height of the streetlight.
Mixture
Pure water is being added to 153 ml of a
solution of 30% hydrochloric acid. How much
pure water should be added to dilute to 13%?
2.3 Linear Inequalities
Sizing – approximate
Max takeoff weight of aircraft
Interstate speed
x  15  9 x  1
x2  2x 1  2x
xy  x 2  y 3  x
23 1
A linear inequality in one variable is an
inequality that can be written in the form:
ax  b 0 , a  0 .
Properties of inequalities
Let a, b, and c represent real numbers
1. a  b and a  c  b  c are equivalent.
2. If c  0 , then a  b and ac  bc are equivalent.
3. If c  0 , then a  b and ac  bc are equivalent.
Solve symbolically
x2
2x  3 
3
Solve symbolically
 34 z  4   4   z  1
Solve graphically
1
x  2  2x 1
2
The daily payment processing for the company
Square grew from $1M in March 2011 to $11M
in March 2012. Model this growth and find
when daily volume was $8.5M or less.
x intercept method
1 x  1 x  2
2
Numerical Solutions
Let cost of manufacturing be C  x   5 x  200
and revenue be expressed by R x   15 x . Profit
is expressed by revenue minus cost.
Boundary number (break even point)
17 18 19 20
x
10 x  200  30  20  10 0
21
10
22
20
23
30
Solve 36  x   5  2 x  0 numerically.
Y1  36  X   5  2 X
x
y
1
18
2
13
3
8
4
3
5
-2
6
-7
7
-12
x
y
4.3
1.5
4.4
1
4.5
.5
4.6
0
4.7
-.5
4.8
-1
4.9
-1.5
Compound inequalities
Speed limit on Interstate 35
 4  5 x  1  21
1 1  2t

2
2
4
Sunset in Boston
In Boston, the sun set at 7PM on the 82nd day
(22 Mar), and at 8PM on the 136th day (15 May).
Find the days when the sun set between 7:15PM
and 7:45PM. (do not include fall dates).
Symbolic
Graphical
x
 1  3
2
8 
3x  1
5
2
5 x  6   2 x  21  x 
2.4 More Modeling with Functions
To model a quantity that is changing at a
constant rate, with f  x   mx  b ,
f(x)=(const rate of chg)x + (initial value)
Model each situation and state the domain:
(a) In 2011, the average cost of attending a public
college was $8200, and it is projected to increase,
on average, by $600 per year until 2014.
(b) A car’s initial speed is 50 mph, then it
begins to slow down at a constant rate of 10
mph each second.
A 100 gallon tank is initially full of water and
being drained at 5 gallons per minute.
formula?
How much water is in the tank after 4 minutes?
CO2 emissions
x miles 240
y pounds 150
360
230
Slopes? Linear?
f x 
680
435
800
510
f 1000 
Piecewise defined functions
First-Class Mail Rates First ounce $0.46
Each additional ounce $0.20
First-Class Mail Rates
Weight not
over
(ounces) Rate
1
0.46
2
0.66
3
0.86
4
1.06
Tornado
1
2

F  x   3
4

5
Domain?
Range?
Fujita Scale F1  F 5
if 40  x  72
if 72  x  112
if 112  x  157
if 157  x  206
if 206  x  260
Housing starts
Year 2000 2005 2011 2012
Homes 1.3 1.7 0.4 0.5
Graph, interpret, continuous? model
 2  x  2000   1.3
if 2000  x  2005
 25

 13
H  x     x  2005  1.7 if 2005  x  2011
 60
1
if 2011  x  2012
10  x  2011  0.4
Evaluating a piecewise-defined function
Piecewise defined function
 x  1,  4  x  2
f x  
 2 x , 2  x  4
Domain?
f  3, f 2, f 4, f 5
Continuous?
Greatest Integer Function
f  x   x , the greatest integer  x
f 3.7, f 2, f  2.3, f  4
Direct variation
y is directly proportional to x, or y varies
directly with x, if there is a non-zero number
such that
y  kx
The number k is called the constant of
proportionality or the constant of variation.
Wages $57.75 for 7 hours work
Suppose T varies directly with x, and that T  33
when x  5. Find T when x  31.
A 12-pound weight is hung on a spring and it
stretches 2 inches.
Find the spring constant (const of var)
How far will it stretch with a 19lb weight on it?
Megabytes needed for y sec of music
x (MB) 0.23 0.49 1.16 1.27 5.00
y (sec) 10.7 22.8 55.2 60.2
46.5 46.5 47.6 47.4
Computing y
for each entry – 3rd row,
x
approximate constant of proportionality.
Using constant of proportionality, how much
music would a FD hold?
2.5 Absolute Value Equations and Inequalities
Absolute value function
 x , x  0
x 
 x , x  0
Primary square root
f x  x  2
f  x   2 x  4
If x  3, then x  3 .
x
x
Absolute Value Equations
For any positive number k,
ax  b  k is equivalent to ax  b   k
3
x  6  15
4
3x  2  5  2
1  2 x  3
2x  5  2
Graphical
Y1  abs2 X  5
Numerical
TblSet – min  4, 0.5
Symbolic
Y2  2
Interstate speed limits: S  55  15
Two absolute values
x  2  1  2x
Absolute Value Inequalities
Let the solutions to ax  b  k be s1 and s2 , with
s1  s2 , and k  0 .
1. ax  b  k is equivalent to s1  x  s2 .
2. ax  b  k is equivalent to x  s1 or x  s2 .
2x  5  6
5 x  3
Absolute Value Inequalities (Alternative
Method)
Let k be a positive number.
1. ax  b  k is equivalent to  k  ax  b  k
2. ax  b  k is equivalent to
ax  b  k or ax  b  k
4  5x  3
 4x  6  2
Error tolerances on iPhone 4s
The iPhone 4s is 4.5” high. Suppose actual
height A of any particular phone has an error
tolerance of 0.005”.
Model and quantify
Ch 3 Quadratic Functions and Equations
3.1 Quadratic Functions and Models
Linear
f  x   ax  b
y  mx  b
Let a, b, and c be real numbers with a  0 . A
function represented by f  x   ax 2  bx  c is a
quadratic function.
Domain is  , leading coefficient is a.
Graph – parabola
Vertex, Axis of Symmetry
a0
a0
1
y  x2
2
y  x2
y  2x 2
f  x   3  22 x
g  x   5  x  3x 2
3
h x   2
x 1
a  0, vertex 1,2 
a  0, vertex  2,5
Standard form
ax 2  bx  c
Vertex form
a x  h   k
2
f  x   2 x  1  2
2
The parabolic graph of f  x   a x  h   k with
a  0 has vertex h, k . Its graph opens upward
when a  0 , and opens downward when a  0 .
2
Vertex form from graph
Converting to standard form
2
f  x   2 x  1  4
Converting to vertex form
2
k 
k

2
x  kx      x  
2
2 
f x  x2  6x  3
1 2
f x  x  x  2
3
2
y  ax 2  bx  c
The vertex of the graph f  x   ax 2  bx  c with
 b  b 
a  0 is the point   , f    .
 2a  2a  
Find the vertex of f  x   1.5 x 2  6 x  4 .
Symbolic, Graphical, Numerical
Note: for axis of symmetry x  a ,
f a  k   f a  k k
Converting to vertex form:
f  x   3x 2  12 x  7
x
Graph g  x   2 x  1  3
2
b
2a
1 2
h x    x  x  2
2
A farmer is fencing a rectangular area for cattle
using the straight portion of a river as one side
of the rectangle. If the farmer has 2400 feet of
fence, find the dimensions of the rectangle that
will give the maximum area for the cattle.
Models and Applications S t   16t 2  v0t  h0
A baseball is hit straight up with an initial
velocity of v0  80 feet per second and leaves
the bat at an initial height of h0  3 feet.
Write a formula for the height of the ball at t
seconds. What is the height of the ball after 2
seconds?
What is the maximum height of the baseball?
Athlete’s heart rate
0
2
4
6
200 150 110 90
8
80
Modeling quadratic data
2
10
3
1
4
-2
5
1
6
10
f  x   a x  h   k
2
3.2 Quadratic Equations and Problem Solving
A quadratic equation in one variable is an
equation that can be written in the form
ax 2  bx  c  0 , where a, b, and c are real
numbers with a  0 .
x2 1  0
 x2  4x  4  0
x2  2x 1  0
2 x 2  2 x  11  1
12 x 2  x  1
x2  x  2  0
Finding the x-intercepts
24 x 2  7 x  6  0
Square Root property
Let k be a nonnegative number. Then the
solutions to the equation x 2  k are given by
x  k .
100  16 x 2  0
Determine if either equation represents a function.
x 2   y  1  4
2
2y 
x y
2
Completing the square
x 2  8x  9  0
2 x 2  8x  7
x 2  kx  d
2
2
x 2  kx  k 2    x  k 2 
The solutions of the quadratic equation
ax 2  bx  c  0 with a  0 are
 b  b 2  4ac
x
2a
3x 2  6 x  2  0
An athlete’s heart rate was modeled in the last
section as 1.875 x 2  30 x  200 . Determine
when the rate was 110 beats per minute.
Quadratic equation - parabola
Discriminant
To determine the number of real solutions to the
quadratic equation ax 2  bx  c  0 , with a  0 ,
evaluate the discriminant b 2  4ac .
b 2  4ac  0
b 2  4ac  0
b 2  4ac  0
b 2  4ac is a perfect square
4 x 2  12 x  9  0
9 x 2  12.6 x  4.41  0
Construction
A box is being constructed by cutting 2 inch
squares from the corners of a rectangular piece
of cardboard that is 6 inches longer than it is
wide. If the box has a volume of 224 cubic
inches, find the dimensions of the cardboard.
Revenue
A company charges $5 for earbud headphones,
but it reduces this cost by 5¢ for each additional
pair ordered (up to 50). If the total price is $95,
how many earbuds were ordered?
Projectile motion
S t   16t 2  v0t  h0
0
2
4
6
8
96 400 576 624 544
Model it, find max height, and impact to ground
3.3 Complex Numbers
Numbers – Natural – Roman Numerals
Zero – represent nothing?
Negative numbers
Rectangle
Debt
y  x2 1
Properties of the imaginary unit i
i   1 i 2  1
Complex numbers
If a  0 , then
 16
2   24
2
a  bi
 a  i a.
3
3 3
Addition and Subtraction
 2  3i   4  6i 
5  7i   8  3i 
 2 8
Multiplication
 5  i 7  9i 
Division
The conjugate of a  bi is a  bi
a  bi 2  5i 6  3i  2  7i 1  i  4i
3  2i
5i
5
Complex # s  Real# s a  bi, b  0  imaginary # s
Powers of i
i8
Solving quadratics
x 2  3x  5  0
1 x 2  17  5 x
2
2 x 2  3
i19
i 203
3.4 Quadratic Inequalities
x2  4
2  x  x2
2 x 2  3x  2  0 2 x 2  3x  2  0 2 x 2  3x  2  0
Safe stopping distance
1 2 11
x  x
12
5
Visual distance limited to 200 feet
f x 
Graphical Y1  X  2 / 12  11X / 5
Y2  200
To solve symbolically:
1) Replace the inequality symbol with “=”
sign, then solve for the boundary values.
2) Separate the real number line into disjoint
intervals using the boundary values.
3) Test a value in each interval.
4) All values will be positive or negative in
each given interval. Pick interval based on
your inequality – your solution set.
2 x 2  5 x  12  0
x2  2  x
3.5 Transformations of Graphs
Vertical shifts
y  x2
y  x2  2
Horizontal Shifts
y  x2
2
y   x  2
Vertical and Horizontal Shifts
Let f be a function and let k be a positive number.
To graph
y  f x  c
Shift y  f  x  by c units
upward
y  f x  c
downward
y  f x  c
left
y  f x  c
right
y x
y  x2
y  x2 4
Find an equation that shifts the graph of
f  x   4 x 2  2 x  1 right 1980 units and upward
50 units.
Translate a circle
The equation of a circle that has radius 3 with
the center at the origin is x 2  y 2  9. Write the
equation that shifts the circle right 4 units and
upward 2 units.
Stretching and Shrinking
y x
y2 x
y  12 x
Vertical Stretching and Shrinking
If the point  x, y  lies on the graph of y  f  x ,
then the point  x, cy  lies on the graph of
y  cf  x . If c  1, the graph of y  cf  x  is a
vertical stretching of the graph of y  f  x ,
whereas if 0  c  1, the graph of y  cf  x  is a
vertical shrinking of the graph of y  f  x .
Horizontal Stretching and Shrinking
If the point  x, y  lies on the graph of y  f  x ,
x 

then the point  , y  lies on the graph of
c 
y  f cx . If c  1, the graph of y  f cx  is a
horizontal shrinking of the graph of y  f  x ,
whereas if 0  c  1, the graph of y  f cx  is a
horizontal stretching of the graph of y  f  x .
Let this graph represent y  f  x .
y  3 f x
1 

y  f  x
2 
Reflection
1. The graph of y   f  x  is a reflection of the
graph of y  f  x  across the x axis.
2. The graph of y  f  x  is a reflection of the
graph of y  f  x  across the y axis.
Y1   X  4  2
Graphing calculator
Y2  Y1
Y3  Y1  X 
f x  x2  2x  3
 f x   x2  2x  3
f  x   x 2  2 x  3
y  2 x  1  3
2
1
y
x
2
y    x  2  1
Video games and animation
Mountain modeled by f  x   0.4 x 2  4
Plane centered on 1,5
f  x   0.4 x 2  4
2
f  x   0.4 x  2  4
2
f  x   0.4 x  4  4