Jim Jack (J²)
MATH 1314 - College Algebra
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1.1Numbers, Data, and Problem Solving
Natural Numbers
Integers
Rational
Irrational
PEMDAS
- Parenthesis
- Exponents
- Multiplication and Division, left to right
- Addition and Subtraction, left to right
Scientific Notation
A real number r is in scientific notation when r is written as , where and n is an integer.
Dist to sun = 93,000,000
Pop of world in 2050 = 9,000,000,000
Light travels in 1 mile =
Mass of the earth – kg
Calculator Exercise
Find the speed the earth travels in space relative to the sun in miles per hour.
Find the volume of a soda can with radius 1.4” and height 5”. Can it hold 16oz?
1.2 Visualizing and Graphing Data
Low temperatures in Minneapolis for six nights
-12 / -4 / -8 / 21 / 18 / 9maximum, minimum, mean, median
A relation is a set of ordered pairs.
Portland rainfall
1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 126.2 / 3.9 / 3.6 / 2.3 / 2.0 / 1.5 / 0.5 / 1.1 / 1.6 / 3.1 / 5.2 / 6.4
Domain=Range=
20 / 20 / 40 / 401.2 / 1.1 / 1.5 / 1.6
Domain=Range=
Portland rainfall
1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 126.2 / 3.9 / 3.6 / 2.3 / 2.0 / 1.5 / 0.5 / 1.1 / 1.6 / 3.1 / 5.2 / 6.4
Distance formula
Distance between two moving cars
100
50
KC50100
Linear Approximation
Midpoint formula
The midpoint of a line segment with endpoints and in the xy plane is
Population of the US
In 1990, the population of the US was 249 million, and by 2020, it was 335 million.
Find the population in 2005.
A circle is the set of points equidistant from a center point.
Graph
Finding the equation of a circle
a)Radius = 4, Center is
b)center with a point on the circle
A line is drawn from the point to the point. State the equation of a circle for which this line is the circle’s diameter.
Completing the square
opt
Find the center and the radius of the circle:
1.3Functions and their representations
Thunder and lightning speed of sound=1050 ft/sec
F(time in sec) computes distance
5 / 10 / 15 / 20 / 251 / 2 / 3 / 4 / 5
Rule: Compute y by dividing x by 5
meaningful input – domain
corresponding output – range
Verbal: Divide x by 5 to obtain y miles
Numerical:
1 / 2 / 3 / 4 / 5 / 6 / 70.2 / 0.4 / 0.6 / 0.8 / 1.0 / 1.2 / 1.4
symbolic: Leonhard Euler
(1707-1783)
graphical:
x / -2 / -1 / 0 / 1 / 2/ 2 / 1 / 0 / 1 / 2
Graph
A function is a relation in which each element in the domain corresponds to exactly one element in the range.
exactly one y function (rule for finding y)
Diagrammatic:
51awAx
102bxBy
153cyCz
204zD
Ordered pairs
Domain
Range
The function f computes the revenue in dollars per unique user for different technology companies. This function is defined by , , ,
DomainRange
Let
Domain
DomainRange
Find domain and range graphically
DomainRange
Air cools at a rate of 3.6º F per 1,000 ft altitude
Figure for , where x is in thousands ft.
Symbolic:
Graphical:
Numerical:
Identifying functions
Vertical line test
Determine if y is a function of x.
,
1.4 Types of Functions and Their Rates of Change
Describe datamake predictions
LinearNon-linearConstant
Linear functions
A function represented by , where a and b are fixed, is a linear function. A linear function has a constant rate of change.
A car is initially located 30 miles N of the Red River on I-35 traveling northbound at 60mph.
0 / 1 / 2 / 3 / 4 / 530 / 90 / 150 / 210 / 270 / 330
f=
A function represented by , where b is a constant (fixed number), is a constant function.
Windspeed at Hilo, HI
May / June / July / Aug / Sept / Oct / Nov / Dec7 / 7 / 7 / 7 / 7 / 7 / 7 / 7
f=
Discrete functionContinuous function
Other constant fcns – Thermostat, speed control
Slope as a rate of change
The slope of the line passing through the points and is
Positive slope Negative slope Zero slope No slope
Find the slope of a line through and
. ,
calculate slope.
Slope is a constant rate of change. Any linear function can be written as ,
is y-int, b is y-coord of intercept.
Wages earned at $9.25/hr
Tuition at $350/credit hr + $650 student fee
Distance traveled by light at 186,000 miles/sec.
The function given by calculates the pounds of released into the atmosphere by a car burning x gallons of gasoline.
Find and interpret the result.
Find the slope of the graph of Pand interpret as a rate of change.
Songs that can be stored on x GB of Ipod memory
20,000
songs
80Gb
Pass thru origin?
Slope?
Rate of change?
Let f be any function. Then any number c for which is called a zero of the function f.
Find formula from a graph
Determine the slope, y-int, and the x-int.
Write a formula for f.
Find the zeros (x-intercepts) of f.
Graph linear function , Identify the slope and the y-intercept.
Non-linear functions
Increasing and Decreasing Functions
Suppose f is a function defined over interval I.
For :
a) f increases on I if
b) f decreases on I if
skip
x / 0 / 1 / 2 / 3 / 4f(x) / 0 / 1 / 4 / 9 / 16
average temperature in each month
height of a child at age 2 through 18
Linear and non-linear dataskip
0 / 5 / 10 / 15 / 20-4 / -2 / 0 / 2 / 4
-3 / 0 / 3 / 6 / 9
5 / 7 / 10 / 14 / 19
0 / 1 / 2 / 3 / 4
11 / 11 / 11 / 11 / 11
0 / 1 / 3 / 6 / 10
3 / 6 / 9 / 12 / 15
Recognizing linear functions
skip
Interval Notation
Average rate of change, non-linear function
rate of change from x= 1to x= 3
rate of change of US pop, 1800 – 1840, 1900 – 1940
Year / Population1800 / 5
1840 / 17
1900 / 76
1940 / 132
The difference quotientopt
Find first, then difference quotient.
Ch 2 Linear Functions and Equations
2.1Equations of lines
Point-slope form
Change in x is
Change in y is
x intercept , y intercept
Slope passing through
Equation of a line passing through ,
Alternate method slope-intercept form
Equation of a line passing through ,
In 2011, Facebook's daily active users were about 460 million, and this number increased to 1 billion in 2015.
Approximate the linear equation of this growth. Interpret the slope of this line.
Find the number of users in 2012, predict for 2019.
Interpreting intercepts and slope
Graph shows the distance an athlete training for a marathon is from home after x hours.
Find and interpret y-int
Find and interpret x-int
Interpret
Slope? Rate of change?
Slope intercept form of line?
Determine x and y intercepts
Equations of horizontal and vertical lines
An equation of a horizontal line with y intercept is . An equation of the vertical line with x intercept is .
Vertical and horizontal lines through
Finding parallel lines
Two lines with slopes and , neither of which are vertical, are parallel if and only if .
Find the equation of a line parallel to the line , passing through .
Finding perpendicular lines
Two lines with slopes and , neither of which are vertical, are perpendicular if and only if their slopes have product , i.e..
Find the equation of a line perpendicular to the line , passing through .
Determining a rectangle
Interpolation and Extrapolation
In 2004, there were about 4.6M digital music album downloads. This amount reached $117.6M in 2014. Model, find value in 2006, 2003.
2003 / 2004 / 2006 / 20144.6 / 117.6
Apple sold 55m iPods in 2008, and 43m in 2011
Cars shipped globally in selected years
Year / 2013 / 2015 / 2017 / 2019Investments / 69 / 75 / 81 / 88
Model the data
Interpret slope
Estimate Investments in 2020
Predicted value is 92 million. Extrap/Interp?
2.2Linear 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
Equivalent – two stmts with same solution set
Contradiction – equation with no solution set
Identity – equation with infinite solution set
A linear equation in one variable is an equation that can be written in the form
(an equation that is not linear is called non-linear)
Addition property
Multiplication property
Solving linear equations symbolically
Solving linear equations symbolically
Eliminating fractions and decimals
The linear function estimates iPod sales (million) during year x. When will sales reach 27 million?
Contradictions, Identities, Conditional Equations
An equation with one expression on each side of the = sign can be solved by the intersection of graphs method.
Using a graphing calculator;
- Set equal to the left side, and equal to the right side of the equation.
- Graph both on one set of coordinates
- Points of intersection are ordered pairs that make both sides true at the same time. These ordered pairs solve the equation.
Skip CDs
The market share of music on CDs held from 1987 to 1998 could be modeled by . During the same period, cassette tape sales could be modeled by . (x=0 indicates year 1987)
Percentages: Write each percentage as a decimal.
45%0.03%420%-1.45%%
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
Problem Solving
- Read problem, understand it. Assign variable to what you must find. Write other quantities in terms of this variable.
- Model it, write the equation. Diagram it if necessary.
- Solve the equation, state solution
- 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%?
2.3Linear Inequalities
Sizing – approximate
Max takeoff weight of aircraft
Interstate speed
A linear inequality in one variable is an inequality that can be written in the form:
, .
Properties of inequalities
Let a, b, and c represent real numbers
1. and are equivalent.
2. If , then and are equivalent.
3. If , then and are equivalent.
Solve symbolically
Solve symbolically
Solve graphically
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
Numerical Solutions
Let cost of manufacturing be and revenue be expressed by . Profit is expressed by revenue minus cost.
Boundary number (break even point)
x / 17 / 18 / 19 / 20 / 21 / 22 / 23/ / / / 0 / 10 / 20 / 30
Solve numerically.
x / 1 / 2 / 3 / 4 / 5 / 6 / 7y / 18 / 13 / 8 / 3 / -2 / -7 / -12
x / 4.3 / 4.4 / 4.5 / 4.6 / 4.7 / 4.8 / 4.9
y / 1.5 / 1 / .5 / 0 / -.5 / -1 / -1.5
Compound inequalities
Speed limit on Interstate 35
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
2.4 More Modeling with Functions
To model a quantity that is changing at a constant rate, with,
f(x)=(const rate of chg)x+ (initial value)
Model each situation and state the domain:
(a) In 2015, the average cost of attending a public college was $9400, and it is projected to increase, on average, by $300 per year until 2018.
(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?
emissions
x miles / 240 / 360 / 680 / 800y pounds / 150 / 230 / 435 / 510
Slopes? Linear?
Piecewise defined functions
First-Class Mail Rates First ounce $0.49
Each additional ounce $0.21
First-Class Mail Rates
Weight not
over
(ounces) Rate
1 0.49
2 0.70Domain?
3 0.91
4 1.12Range?
TornadoFujita Scale
Housing starts
Homes / 1.3 / 1.7 / 0.4 / 1.1
Graph, interpret, continuous? model
Evaluating a piecewise-defined function
Piecewise defined function
Domain?
Continuous?
Greatest Integer Functionopt
, the greatest integer x
Direct variation
y is directly proportional to x, or y varies directly with x, if there is a non-zero number such that
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 when . Find T when .
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
y (sec) / 10.7 / 22.8 / 55.2 / 60.2
46.5 / 46.5 / 47.6 / 47.4
Computing for each entry – 3rd row, 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
Primary square root
If , then .
Identify all intercepts of
Where is ? ? ?
Increasing? Decreasing?
Domain?
Range?
Absolute Value Equations
For any positive number k,
is equivalent to
Graphical
NumericalTblSet –
Symbolic
Interstate speed limits:
Two absolute values
Absolute Value Inequalities
Let the solutions to be and , with , and .
1. is equivalent to .
2. is equivalent to .
Solve numerically and graphically
Solve symbolically
Absolute Value Inequalities (Alternative Method)
Let k be a positive number.
1. is equivalent to
2. is equivalent to
Error tolerances on iPhone 6
The iPhone 6 is 5.44” 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
3.1Quadratic Functions and Models
Linear
Let a, b, and c be real numbers with . A function represented by is a quadratic function.
Domain is , leading coefficient is a.
Graph – parabola
Vertex, Axis of Symmetry
Standard form
Vertex form
The parabolic graph of with has vertex . Its graph opens upward when , and opens downward when .
Vertex form from graph
Converting to standard form
Converting to vertex form
The vertex of the graph with is the point.
Find the vertex of .
Symbolic, Graphical, Numerical
Note: for axis of symmetry ,
Converting to vertex form:
Graph
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
A baseball is hit straight up with an initial velocity of feet per second and leaves the bat at an initial height of 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 / 8200 / 150 / 110 / 90 / 80
Modeling quadratic data
2 / 3 / 4 / 5 / 610 / 1 / -2 / 1 / 10
3.2 Quadratic Equations and Problem Solving
A quadratic equation in one variable is an equation that can be written in the form , where a, b, and c are real numbers with .
Finding the x-intercepts and the y-intercept
(Factoring assistance in Section R.4)
Square Root property
Let k be a nonnegative number. Then the solutions to the equation are given by .
Determine if either equation represents a function.
Completing the square
The solutions of the quadratic equation with are
An athlete’s heart rate was modeled in the last section as . Determine when the rate was 110 beats per minute.
Discriminant
To determine the number of real solutions to the quadratic equation , with , evaluate the discriminant .
is a perfect squarequadratic factors
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
0 / 2 / 4 / 6 / 896 / 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
Properties of the imaginary unit i
Complex numbers
If , then .
Addition and Subtraction
Multiplication
Divisionopt
The conjugate of is
Powers of i
Solving quadratics
3.4 Quadratic Inequalities
Safe stopping distance
Visual distance limited to 200 feet
Graphical
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.
3.5 Transformations of Graphs
Vertical shifts
Horizontal Shifts
Vertical and Horizontal Shifts
Let f be a function and let k be a positive number.
To graphShift by c units
upward
downward
left
right
Find an equation that shifts the graph of 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 . Write the equation that shifts the circle right 4 units and upward 2 units.
Stretching and Shrinking
Vertical Stretching and Shrinking
If the point lies on the graph of , then the point lies on the graph of . If , the graph of is a vertical stretching of the graph of , whereas if , the graph of is a vertical shrinking of the graph of .
Horizontal Stretching and Shrinking
If the point lies on the graph of , then the point lies on the graph of . If , the graph of is a horizontal shrinking of the graph of , whereas if , the graph of is a horizontal stretching of the graph of .
Let this graph represent .
Reflection
1. The graph of is a reflection of the graph of across the x axis.
2. The graph of is a reflection of the graph of across the y axis.
Graphing calculator
Video games and animation
Mountain modeled by
Plane centered on