Sect 1.1

1
x coordinates
y coordinates
Compare all the x coordinates,
The set is not a function, just a relation.
repeats.
Compare all the x coordinates, no repeats. The
set is a function.
2
Compare all the x coordinates in the domain, only
one corresponding arrow on each x coordinate.
The set is a function.
Compare all the x coordinates in the domain, 8
has two corresponding arrows. Repeats
The set is not a function, just a relation.
3
When determining if a graph is a function, we
will use the Vertical Line Test.
Use your pencil as a Vertical Line and place it
at the left side of the graph.
Slide the pencil to the right and see if it
touches the graph ONLY ONCE. If it does it is a
FUNCTION.
FUNCTION.
Use your pencil as a Vertical Line and place it
at the left side of the graph.
The Vertical Line crosses the graph in 2 or more
locations, therefore this graph is just a
RELATION.
4
Not multiplication!
y coordinates
input
output
y f(x)
Dependent Variable
Independent Variable
y 3(4) 7
y 12 7
y 19
The work is the same!
f(4) 3(4) 7
f(4) 12 7
f(4) 19
5
Put ( )s around every x.
Substitute -6 for every x.
10
Simplify by Order of Operations.
FOIL and distribute
Combine Like Terms, CLT.
6
Remember h(x) y
???
h(x) y
h(x) y
h(x) y
h(3) 2
h(2) 1
h(0) is not possible! Zero is not in the
Domain. h(0) undefined
y
(3, 5)
y
y
Find the point when x 3
Find the point when x -2
Find the point when x 0
(-2, 1)
j(3) 5
j(-2) 1
0
j(0) -1
3
-2
(0, -1)
7
Every x coordinate from 3 to 6
y
(6, 5)
(3, 5)
5
Find the point when y 3
Find the point when y 1
3
(2, 3)
(-4, 1)
x 2
x -4, -2 1
1
(1, 1)
(-2, 1)
-3
(?, 3)
(-?, -3)
Find the point when y 5
Find the point when y -3
3 lt x lt 6
j(x) -3 is not possible! -3 is not in the
Range.
3, 6 interval notation
8
Domain Find the smallest x coordinate to the
largest x coordinate.
7
5
Domain -7 lt x lt 6 or -7, 6
3
-7
Range Find the smallest y coordinate to the
largest y coordinate.
6
The first set of y coordinates are -4 lt y lt 3
or (-4, 3). Notice that we started and ended at
open circles.
-4
The second set of y coordinates are 5 lt y lt 7
or 5, 7
Open circles mean that the point doesnt exist
and the closed circle means that the point is
there. x -3 at this locationas long as we can
touch the graph the x coordinates are there and
continuous.
Range -4 lt y lt 3 or 5 lt y lt 7
(-4, 3) U 5, 7
9
Domain Find the smallest x coordinate to the
largest x coordinate.
Domain x gt -4 or -4, oo)
Range Find the smallest y coordinate to the
largest y coordinate.
-4
Range y gt -7 or -7, oo)
-7
10
Domain Find the smallest x coordinate to the
largest x coordinate.
4
Domain -8 lt x lt 8 or -8, 8
1
Range Find the smallest y coordinate to the
largest y coordinate.
-8
8
-1
The y coordinates are not connected or
consistent, therefore we list them
separately. Range -1, 1, 4
When given the function in set notation, list the
x and y coordinates separately.
Domain -1, 1, 2, 3, 4, 5
Range 1, 2, 3, 4, 7, 8
11
Find the domain of the functions.
When finding the domain of functions in equation
form we will ask ourselves the following
questions. Will the function work when the x is
a negative?, . a zero?, a positive? If
the answers are 3 yess, then the domain is all
real numbers. If there is a no, then there is a
domain restriction we need to find.
Can I multiply 4 by a negative?, a zero?, a
positive? and then add 2 to the product?
ALL Yes!
Domain is ALL REAL NUMBERS
Can I square a negative?, a zero?, a positive?
and then add 2 to the value?
ALL Yes!
Domain is ALL REAL NUMBERS
If I square a negative?, a zero?, a positive? I
should be able to raise them to any power!
ALL Yes!
Domain is ALL REAL NUMBERS
12
Find the domain of the functions.
Adding and subtracting always is a YesCan I
divide by a negative?, a zero?, a positive?
NO! Cant divide by ZERO! Set the denominator
equal to zero and solve for x to find the
restriction.
Domain is ALL REAL NUMBERS, except 1
Can I take the absolute value of a negative?, a
zero?, a positive?
ALL Yes!
Domain is ALL REAL NUMBERS
We have a fraction again, set the bottom equal to
zero and solve for x.
Domain is ALL REAL NUMBERS, except for -3 and 3.
13
We have a fraction again, set the bottom equal to
zero and solve for x by factoring.
Domain is ALL REAL NUMBERS, except for -8 and 2.
We have a fraction again, set the bottom equal to
zero and solve for x by factoring.
Domain is ALL REAL NUMBERS, except for -3, 0 and
4.
14
Domain Restrictions
5 lt -5, FALSE
-5 lt 5 lt 3 , FALSE
5 gt 3, TRUE
x 5, test it in the domain restrictions to see
which one is true! Substitute the 5 into that
function.
x 3, and 3 gt 3. Substitute 3 into the third
function.
x -7, and -7 lt -5. Substitute -7 into the
first function.
x -5, and -5 lt -5 lt 3. Substitute -5 into the
second function.
15
Cubic Func.
16
(0, 6)
rise run
m slope
down 5
b y-int (0, b)
right 2
starting point
y-int (0, 6)
down 5
directions
-5 2
m
right 2
17
right 3
up 1
right 3
up 1
right 3
point (x1, y1)
up 1
left 3
rise run
down 1
m slope
(-3, 4)
starting point
y-int (-3, 4)
directions
1 3
m
Or in reverse
18
A, B, and C are integers.
To graph find x and y intercepts
???
To find the y intercept the x coordinate is
zero! (0, y)
To find the x intercept the y coordinate is
zero! ( x, 0)
Doesnt fit, but that is okwe can use the slope!
19
Notice that there is no y variable in the
equation. This means we cant cross the y axis!
Must be a VERTICAL LINE at x 6
rise 0
m slope undefined
Notice that there is no x variable in the
equation. This means we cant cross the x axis!
Must be a HORIZONTAL LINE at y - 4
0 run
m slope 0
20
To graph find x and y intercepts. We can see that
3 will divide into -9 evenly, but 5 wont. So we
should find the x intercept and the slope to
graph this line.
To find the x intercept the y coordinate is
zero! ( x, 0)
Find the slope!
21
Write the equation of a line that contains the
points ( 3, 8 ) and ( 5, -1 ).
Find the slope first.
Next use the point-slope form to write the
equation.
Convert to y mx b.
Yellow TAXI Cab Co. charges a 10 pick-up fee and
charges 1.25 for each mile. Write a cost
function, C(m) that is dependent on the miles, m,
driven.
Rememberfunctions are equal to y. y C(m).
Use y mx b.
The slope is the same as rate! The y intercept
(b) is the starting point or initial cost.
The 10 pick-up fee is a one time charge or
initial cost. b 10
The 1.25 for each mile is a rate. m 1.25
Replace y with C(m) and x with m.
22
In the year 2000, the life expectancy of females
was 83.5. In 2004, it was 86.5. Write a linear
function E(t) where t is the number of years
after 2000 and E(t) is the life expectancy in t
years. Estimate the life expectancy in the year
2009. Estimate the when the life expectancy will
be 94.
Looks difficult only because of all the words!
Understand the data given to write the equation
of a line!
This looks like points (x, y) (t, E(t))
Year of years after 2000 (t) Age E(t)
2000 0
83.5 2004 4
86.5
(0, 83.5)
(4, 86.5)
We are back to the first problem we did for
writing the equation of a line. Use y mx b
because we are working with functions and (0,
83.5) is the y intercept.b is 83.5.
Find the slope between the points.
Estimate the life expectancy in the year 2009.
Estimate the when the life expectancy will be 94.
14 years past the year 2000, 2014.
23
In the year 2003, a certain college had 3450
students. In the year 2008, the college had 4100
students. Write a linear function P(t) where t
is the number of years after 2000 and P(t) is the
population of the college. Estimate the
population in the year 2012. Estimate the year
when the population will reach 5400.
Understand the data given to write the equation
of a line!
Year of years after 2000 (t) Students
P(t) 2003 3
3450 2008 8
4100
Points (x, y) (t, P(t))
(3, 3450)
(8, 4100)
Use y mx b because we are working with
functions, but this time we will have to solve
for b.
Find the slope between the points.
Plug in a point, (8, 4100).
Estimate the population in the year 2012.
Estimate the year when the population will reach
5400.
18 years past the year 2000 is the year 2018.
24
Same Line
Yes, (-4, 7) is a solution.
25
Find the solution to the system by graphing.
( 1, 2 )
Solution is ___________.
26
Find the solution to the system by graphing.
Convert to y mx b
The slopes are the same and the y-intercepts are
different.
Solution is ___________.
No Solution
27
Find the solution to the system by graphing.
Divide everything by 4.
Same LINE! Infinite Solutions, but not the final
answer.
Convert to y mx b
Solution is ___________.
Answer must be written as a point.
28
Section 8.2. Solving linear systems by
SUBSTITUTION ELIMINATION.
Solve for x y.
Substitution Method. 1. Choose an equation and
get x or y by itself. 2. Substitute step 1
equation into the second equation. 3. Solve for
the remaining variable. 4 Substitute this
answer into the step 1 equation.
Step 1
Step 2
Step 3
Step 4
Is the intersection point and solution.
29
Solve the system for x and y.
False Stmt. No Solution
True Stmt. Infinite Solutions But not done!
Answer should be a point ( x, mx b )
30
Sect 8.1 Systems of Linear Equations
Solve for x y.
Step 1
Elimination Method. 1. Choose variable to
cancel out. Look for opposite signs. 2. Add
the equations together to cancel. 3. Solve for
the remaining variable. 4 Substitute this
answer into either equation in the step 1
equations.
The y-terms are opposite signs. Multiply the
first equation by 3 and the second equation by 4.
Step 2

Step 3
Step 4
(3, 2) is the intersection point and solution.
31
Solve the system for x and y.
The x-terms are the smallest and easiest to
cancel. To determine which factor will be
negative check the y-terms
The y-terms are multiples of 4. Multiply the 1st
equation by -2 to make opposites.

Add equations together.

32
Solve the system for x and y.
Remove fractions by multiply by the LCD and
decimals by multiply by 10s



33
Solve the system for x and y.
True Stmt. Infinite Solutions Solve for y.
False Stmt. No Solution
Answer should be a point ( x, mx b )
34
Total-Relationship Systems.
In 2008, there were 746 species of plants that
were considered threatened or endangered. The
number considered threatened was 4 less than a
fourth of the number considered endangered. How
many plants are considered threatened and
endangered in 2008?
Find the two unknowns from the question and
determine their TOTAL. Always read the question
sentence first.
T E 746
T How many threatened? E How many endangered?
Now read through the details to find the
RELATIONSHIP between the two variables.
T ¼( E ) 4
Substitution Method
35
Total-Relationship System.
The sum is 90 o
Two angles are complementary. One angle is 12o
less than twice the other. Find the measure of
the two angles.
Find the two unknown from the question and
determine their TOTAL. Always read the question
sentence first.
A B 90
A First angle B Second angle
Now read through the details to find the
RELATIONSHIP between the two variables.
A 2( B ) 12
Substitution Method
If the two angles are supplementary, then the
sum is 180 o
36
Total-Rate Systems.
A jewelry designer purchased 80 beads for a total
of 39. Some of the beads were silver beads that
cost 40 cents each and the rest were gold beads
that cost 65 cents each. How many of each type
did the designer buy?
Find the two unknown from the question and
determine their TOTAL. Always read the question
sentence first.
G S 80
G How many Gold beads? S How many Silver
beads?
Now read through the details to find the RATE on
each variable. Multiply the rates to the
variables and set equal to total cost.
0.65G 0.40S 39.00
Elimination Method Cancel smallest variable term.
37
Total-Rate Systems.
Janes student loans total 9,600. She has a PLUS
loan at 8.5 and a Stafford loan at 6.8 simple
interest. In one year, she was charged 729.30
in simple interest. How much was each loan?
Find the two unknown from the question and
determine their TOTAL. Always read the question
sentence first.
P S 9600
P How much was the PLUS loan? S How much was
the Stafford loan?
Now read through the details to find the RATE on
each variable. Multiply the rates to the
variables and set equal to the total
cost. Percent must be changed to a decimal!
0.085P 0.068S 729.30
Elimination Method Cancel smallest variable term.
38
Total-Rate Systems.
A child ticket costs 3 and an adult ticket costs
5 at an afternoon movie. 300 tickets were sold
for 1,150. How many of each type of ticket were
purchased?
Find the two unknown from the question and
determine their TOTAL. Always read the question
sentence first.
C A 300
C How many Child tickets? A How many Adult
tickets?
3C 5A 1150
Now read through the details to find the RATE of
each variable. Multiply the rates to the variable
and set equal to total cost.
Elimination Method Cancel smallest variable term.
39
Total-Mixture Systems.
Cashews cost 4 per pound and Walnuts cost 10
per pound. How much of each type should be used
to make a 50 pound mixture that sells for 5.80
per pound?
Find the two unknown from the question and
determine their TOTAL. Always read the question
sentence first.
C How many pounds of Cashews? W How many
pounds of Walnuts?
C W 50
Now read through the details to find the RATE of
each variable and TOTAL. Multiply the rates to
the variable and TOTAL.
4C 10W 5.80(50)
Elimination Method Cancel smallest variable term.
40
Total-Mixture Systems.
Home Depot carries two brands of liquid
fertilizers containing nitrogen and water.
Gentle Grow is 3 nitrogen while Super Grow is 8
nitrogen. Home Depot needs to combine the two
types of solution to fill a customers order that
requested 90L of fertilizer that is 6 nitrogen.
How much of each brand should be used to fill the
order?
Find the two unknown from the question and
determine their TOTAL. Always read the question
sentence first.
G How many liters of Gentle Grow? S How many
liters of Super Grow?
G S 90
Now read through the details to find the RATE of
each variable and TOTAL. Multiply the rates to
the variable and TOTAL.
0.03G 0.08S 0.06(90)
Elimination Method Cancel smallest variable term.
41
Distance Systems.
A jet flies 4 hours west with a 60 mph tailwind.
Returning against the same wind, the jet takes 5
hours. What is the speed of the jet with no wind?
D r t
With wind Against wind
Substitution Method
540 mph in no wind.
42
Distance Systems.
A freight train leaves Chicago heading to Denver
at a speed of 40 mph. Two hours later an Amtrak
train leaves Chicago bound for Denver at a speed
of 60 mph. How far will the trains travel until
the Amtrak passes the freight train?
D r t
Freight train Amtrak
Substitution Method
They will travel 240 miles.