Math 6H ( Periods 3, 6, & 7)

Word Problems 6-3 & 6-4
Read-- and read read your word problems. Really determine what the question is asking...

When in doubt, draw it out!!

Sally & Bill prepared 40 sandwiches for a picnic with 14 friends. If the sandwiches were divided equally among all of them, how many sandwiches would each person receive?
Wait, think... there are 16 total people because you must count Sally & Bill
so 40/16 = 2 1/2 sandwiches

A class has 48 minutes available to hear reports for 15 students. If the time is to be divided equally, how long may each student take?

48/ 15 = 3 1/5 minutes


An assembly period has 50 minutes for speeches by 4 candidates for class president. If two minutes are allowed for each introduction and the remaining time is to be divide equally, how long may each candidate speak?

4 introductions so 2 X 4 = 8 50 -8 = 42 minutes to be divided equally

42/ 4 = 10 1/2 minutes each


A fence is to be put up along a side of a lot measuring 114 feet. Nineteen fence posts are to be used-- one at each end and 17 equally spaced between ends. What is the distance between centers of two consecutive fence posts?


This is a Must DRAW OUT


I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I

count the spaces between and you will find that there are 18 spaces between the posts so 114/18 = 6 1/3 ft


A play consists of 3 acts of equal length with 2 twenty minute intermissions between acts. (this means there is one between the first and the second act and another between the second and the third act) If the whole production takes 3 hours, how long is each act in minutes?

When in doubt-- draw it out


l_______________l_____l_______________l_____l_______________l

First change 3 hours to 180 minutes

then deduct the two twenty minute intermissions 180 - 40 = 140

140 minutes left for the three equal acts

140/3 = 46 2/3 min



An hour TV program consists of four segments of equal length. There are 3 on-minute commercials between EACH segment. There is also a 2 minute intro and a 2 minute closing. How long is each segment?


When in doubt-- draw it out


l______l__________l__l__l__l__________l__l__l__l__________l______l


there are 4 minutes for into and closing and 3(3) minutes of commercials 4 + 9 = 13 minutes

60 -13 = 47 47/ 4 = 11 3/4 minutes



A table top is in the shape of a square 45 inches on each side. A chessboard is inlaid so that a border of 3 inches exists on all four sides. How wide is each of the 8 rows of squares of chessboard?



When in doubt... draw it out


You will see that there is a border of 3 inches all around so you need to deduct 3 inches from one side and 3 inches from the other or 6 inches


45-6 = 39. Now the chessboard has 8 rows so divide

39/8 = 4 7/8 inches


The Bears won 11 out of 16 games. The Eagles won 17 out of 24 games. Which team won a greater fraction of its games?

Compare

11/16 or 17/24
WAIT... not those two fractions again!!

The Eagles...
Did you have to change to a common denominator??? I hope not!!


There were 10 sunny days out of 28 in February and 12 sunny days out of 30 in November. (Anyone remember the poem...or the way to check with your knuckles???)

Which month had the higher fraction of sunny days?

10/28 12/30 I might simplify first but I can tell by comparing that November had more sunny days


What method did you use???

Math 6H ( Periods 3, 6, & 7)

Comparing Fractions 6-4
When two fractions have equal denominators it is easy to tell which of the fractions is greater.
We simply compare their numerators.
3/11 < 5/11 since 3< 5 If the fractions have different denominators, there are a variety of methods to consider. We could find a common denominator, which we will need to do when we add or subtract fractions... but when comparing let's try other methods... Take 2/3 and 4/5 Comparing Fractions


Or compare 5/6 and 7/9

again this time you would multiply
5(9) = 45 and 7(6) = 42
so 5/6 > 7/9

Also if the numerator is the same
2/3, 2/7, 2/9, 2/11, 2/21, 2/35

The larger the denominator the smaller the fractions so to list in order from least to greatest start with the largest number in the denominator!!

and if you have fractions with the numerator just one away from the denominator
such as 3/4, 5/6, 7/8, 9/10, 23/24, 45/46
the smallest fraction will be the one with the smallest numbers
3/4 is the smallest fraction and that list is in order from least to greatest!!

What if you need to name a fraction between two fraction 1/6 and 3/8
you could find the LCD
1/6 = 4/24 and
3/8 = 9/24
so you could state
5/24, 6/24 ( but that is really 1/4), 7/24, or 8/24 ( but that is really 1/3.
There are actually an infinite number of fractions... these are only 4 of them

What if you need to find a fraction between 3/7 and 4/7
sometimes you need to change the denominators just to realize that there really are other fractions between
for instance, 3/7 = 6/14 and 4/7 = 8/ 14 so doesn't 7/14 ( or actually 1/2) work!!
... and that's just one of the fractions!!

If n > 0
Then
if a < b a/n < b/n Think about this one!! Plug in some numbers and see what happens and if a < b, then n/a > n/b
Again, plug in some numbers and see what happens!!

If a/b and c/d are fractions and if ad > bc, which fraction is greater,
a/b or c/d ?
Post your answer below in the comments for extra credit. Make sure to give your reasoning for your answer.


Ordering or comparing fractions:

Different ways:

I. Benchmarks - 0, 1/4, 1/2, 3/4, and 1 (using your gut feeling)
How do you figure out which benchmark to use?

When the numerator is close to the denominator, the fraction is approaching 1

(Ex: 9/11)

When you double the numerator and it's close to the denominator, the fraction is close to 1/2
(Ex: 4/9)

When the numerator is very far from the denominator, the fraction is approaching zero
(Ex: 1/8)

Also, if one number is improper or mixed number and other is a proper fraction,
then obviously the number greater than 1 will be bigger!

II. LCD - give them all the same denominator using the LCM as the LCD


III. Use cross multiplication when comparing two... do it several times when comparing a list of fractions

IV. Change them to decimals ( works well if you are great at decimals-- but I want you to become GREAT at fractions!!)

Algebra Period 4

Find the Equation of a Line
: 7-6

1) We know how to GRAPH a line by 3 points where we decide what to plug in and chug

Usually, we just try 0, 1, 2 first


2) We know how to GRAPH a line by intercepts...we plug in zero for y and x and chug

This works really well when the line is in STANDARD form and the coefficients are factors
 of the constant on the other side of the equation.


3) We know how to GRAPH a line by using slope-intercept form:


We isolate y on one side

We read the y intercept (the b which is the constant on the other side)

We graph that value on the y axis

We COUNT to the next point by reading the slope, the coefficient of the x


The slope should be read as the y value (+up or -down)
and then the x value (+ right or - left)


If the slope is not a fraction, make it a fraction by putting the integer over 1



TODAY WE'RE NOT LEARNING TO GRAPH ANOTHER WAY...WE'RE LEARNING HOW TO FIND THE EQUATION OF A LINE

We usually use the slope-intercept form of the line as our "template"

We know that y = mx + b so


we can substitute in what we know (what the problem gives us)
 and solve for whatever we're missing!


[The problem will give you enough information to know the slope or know the y intercept! Then you'll solve for the missing item.]



It helps to memorize this little rhyme


Oh mystery line,
What could you be?

If I could just find you,

y = mx + b


If I could find m

And I could find b.

Then I could put it together

And I would see:

y = mx + b


The rhyme has 3 steps and usually you will have 3 steps or questions to ask yourself:


1) Do I have the slope (m)? If not, find it by using the slope formula or counting it if you have the graph


2) Do I have the y intercept (b)? If not, find it by plugging in a point and the slope and solving for b or if you have the graph, just read it on the y axis.


3) Don't forget to put it all together in one equation at the end!



THERE ARE 5 CASES THAT YOUR BOOK INCLUDES:


First case:
You're given the slope and the y intercept
 (easiest case) 
m = 3/2 and b = -7/5


Just plug in to the generic slope intercept equation: y = 3/2 x - 7/5



Second case:
You're given a point and the slope and need to find the intercept (b)


(3, 1) is a point on the line and m = 2

Plug in the point and the slope and solve for b


1 = 2(3) + b
1 = 6 + b


b = -5


Now put it altogether with the given slope and the intercept you just found:


y = 2x -5



Third case:
You're given a point and the y intercept and need to find the slope


(3, 1) is still a point on the line, but this time you know b = 2


Plug in the point and the y intercept and solve for slope


1 = 3m + 2
-1 = 3m
m = -1/3


Now put it altogether with the given intercept and the slope you just found:


y = -1/3 x + 2



Fourth case:
You're given 2 points and need to find the slope and the intercept

(1 , 3) and (-2 , -3) are 2 points on the line


You need to first find the slope using the formula:

m = change in y / change in x = 3 - (-3)/ 1 - (-2) = 6/3 = 2


Now plug the slope in with one (you pick the easiest!) of the points and find the intercept, b


3 = 2(1) + b

3 = 2 + b
b = 1


Finally, put it all together:
 y = 2x + 1



Fifth case:
You have a graph of a line and need to determine the equation


Look at the graph and find 2 easy points to use to find the slope (make sure they are integers!)


(If the y intercept is not an integer, then follow fourth case completely)


Put the information together in y = mx + b form

Math 6H (Period 3, 6 & 7)

Fractions & Mixed Numbers 6-3

1/2 + 1/2 + 1/2 = 3/2

A fraction whose numerator is greater than or equal to its denominator is called an improper fraction.
Every improper fractions is greater than 1
A proper fraction is a fraction whose numerator is less than its denominator.
Thus, a proper fraction is always between 0 and 1

1/4, 2/3, 5/9. 10/12 17/18 are all proper fractions


5/2, 8/3, 18/15, 12/5 are all improper fractions


You can express any improper fraction as the sum of a whole number and a fraction
a number such as 1 1/2 is called a mixed number

If the fractional part of a mixed number is a proper fraction in lowest terms, the mixed number is said to be in simple form.


To change an improper fraction into a mixed number in simple form, divide the numerator by the denominator and express the remainder as a fraction.
14/3 = 4 2/3
30/4 = 7 2/4 = 7 1/2


To change a mixed number to an improper fraction rewrite the whole number part as a fraction with the same denominator as the fraction part and add together.

or multiply the denominator by the whole number part and add the fractional part to that...
In class I showed the circle shortcut. If you were absent, check with a friend or ask me in class!!

2 5/6 =

(2 x 6) + 5
6
=17/6


Practice these:

785 ÷ 3

852÷ 5

3751÷ 16

98001÷231


post your answers below in the comments for extra credit !!

Math 6H (Period 3, 6, & 7)

Equivalent Fractions 6-2


We drew the four number lines from Page 182 and noticed that 1/2, 2/4, 3/6, and 4/8 all were at the midpoints of the segment from 0 to 1. They all denoted the same number and are called equivalent fractions.

If you multiply the numerator and the denominator by the same number the results will be a fraction that is equivalent to the original fraction

1/2 = 1 x 3/2 x 3 = 3/6

It works for division as well
4/8 = 4 ÷ 4 / 4 ÷ 8 = 1/2

So we can generalize and see the following properties
For any whole numbers a, b, c, with b not equal to zero and c not equal to zero

a/b = a x c/ b x c and
a/b = a ÷ c / b ÷c


Find a fraction equivalent to 2/3 with a denominator of 12
we want a number such that 2/3 = n/12
You could look at this and say
" What do I do to 3 to get it to be 12?
Multiply by 4
so you multiply 2 by 4 and get 8 so
8/12 is an equivalent fraction


A fraction is in lowest terms if its numerator and denominator are relatively prime-- That is if their GCF is 1

3/4, 2/7, and 3/5 are in lowest terms.
They are simplified
You can write a fraction in lowest terms by dividing the numerator and denominator by their GCF.


Write 12/18 is lowest terms
The GCF (12 and 18) = 6

so 12/18 = 12÷ 6 / 18 ÷ 6 = 2/3

Find two fractions with the same denominator that are equivalent to 7/8 and 5/12
This time you need to find the least common multiple of the denominators!! or the LCD
Using the box method from Chapter 5, we find that the LCM (8, 12 ) = 24

7/8 = 7 X 3 / 8 X 3 = 21/24
and
5/12 = 5 X 2 / 12 X 2 = 10/24


When finding equations such as
3/5 = n/15 we noticed we could multiply the numerator of the first fraction by the denominator of the second fraction and set that equal to the denominator of the first fraction times the numerator of the second... or

3(15) = 5n now we have a one step equation

If we divide both sides by 5 we can isolate the variable n and solve...
3(15)/ 5 = n
9 = n

We found we could generalize

If a/b = c/d then ad = bc

Algebra (Period 4)

Equations and Slope 7-5



Chapter 7-5: SLOPE INTERCEPT FORM



WE HAVE COVERED GRAPHING
 THE FOLLOWING 2 WAYS:


3 random x points using a table (Ch 7-2)


Most pick 0, 1, 2 and just plug and chug to find the y value


This works well when you don't end up with fraction answers!



EXAMPLE: -3y =-2x - 6


x y

0 2

3 4

-3 0
Then we learned a second way to graph:


Graph the 2 special points that are the y and x intercepts (Ch 7-3)


x intercept: where the line intersects the x axis


y intercept: where the line intersects the y axis


If it's in standard form, this way works great if both the x and y coefficients
 are factors of the constant on the other side of the equal sign.
EXAMPLE: 2x - 3y = -6


If x = 0, y = 2 so the y intercept is (0, 2)


If y = 0, x = -3 so the x intercept is (-3, 0)


x y

0 2

-3 0


The x intercept is the one that has the x value (y is 0)


The y intercept is the one that has the y value (x is 0)



TODAY, WE WILL COVER THE MOST USED FORM OF A LINEAR EQUATION:

THE SLOPE INTERCEPT FORM




You must restate the equation to get it into the following format:


y = mx + b


where m = slope and b = y intercept (where the line hits the y axis)
All you do is solve the equation for "y" meaning isolate the y on one side of the equal sign!
Look at the example I gave you above:

-3y =-2x - 6
If you tried the x points 0, 1, 2, only 0 gave an integer for y.
All the other y values were fractions.
The slope intercept form provides the explanation on why.
RESTATE -3y =-2x - 6 INTO SLOPE INTERCEPT FORM:
Divide both sides by -3:
y = 2/3 x + 2


Look at the coefficient for x. WHAT X VALUES WILL GIVE YOU INTEGER ANSWERS FOR Y????
They need to be multiples of 3!



Now look at the graph of y = 2/3 x + 2
Notice that the + 2 at the end is the y intercept! (without doing any work!)
Use the counting method for slope on your graph.
You should have counted: UP 2, RIGHT 3
The slope therefore is 2/3.

Look at the equation.

It told you 2/3 without any work!



Graph when line is in Slope Intercept Form:


So if you have the slope-intercept form of the equation, it's really easy to graph the line:


1) Graph the intercept on the y axis (that's the +/- constant at the end)


2) "Count" the next point by using the slope or x coefficient as a FRACTION
For the equation y = 2/3 x + 2
) Put a dot at (0, 2)

3) From (0,2) count up 2 and over to the right 3 to find the next coordinate (3,4)


Remember, slope is y over x, so the numerator is the y change and the denominator is the x change!


If it's positive, you're counting up (positive) and to the right (positive)
or you can count down (negative) and to the left (negative) because 2 negatives make a positive.



If it's negative, you're counting down (negative) and to the right (positive)
or you can count up (positive) and to the left (negative) because you would have a positive and negative = negative




If you're given the slope and the y intercept,
you can write the equation of any line!


Just use: y = mx + b

EXAMPLE: m = -2/3 and b = -12


The line would be y = -2/3 x - 12



Restate Standard Form to Slope Intercept Form:


Another example: 3x + 4y = 10 is the STANDARD FORM of a line
(x and y on the same side, x is positive, no fractions, constant alone)

Restate into Slope Intercept Form:


Solve for y

first subtract 3x from both sides: 
4y = -3x + 10

Now divide both sides by 4:


y = -3/4 x + 10/4 or
y = -3/4 x + 5/2


The slope is the coefficient of the x
 m = -3/4 (so you're sliding down at a little less than a 45 degree angle)


The y intercept is the constant
b = 5/2 (so the line crosses the y axis at 2 1/2)
Notice that the "b" is a fraction.

When this happens, the slope intercept form may not be the best form to graph the line.

You must start at 2 1/2 on the y axis and count down 3, right 4. THAT'S HARD!
Neither 3 nor 4 is a factor of the constant 10 so intercepts will be fractions.

We need the x term to end up with 1/2 so that when we add that to the b of 5/2, we'll get an integer!
So let's make x = 2 because that will cross cancel with the -3/4 slope to halves:
y = (-3/4)(2) + 5/2 = -3/2 + 5/2 = 2/2 = 1
So we found a coordinate that has just integers: (2, 1)


Now graph that point.


Now count the slope from that point (instead of from the y intercept)!

Pre Algebra (Period 1)

Proportions 6-2

A proportion = 2 equal ratios (2 equivalent fractions)

Solve using equivalent fractions or

Cross multiplication and then a one-step equation

(see if you can simplify the fractions before multiplying)

Example: Solve the proportion for y:


4/3 = y
/21

EQUIVALENT FRACTION APPROACH:
Multiply both top and bottom by 7, y = 28



CROSS PRODUCTS APPROACH:

You'll get 3y = (21)(4)
Now divide each side by 3.
Do this before multiplying on the right side!
Why? Because a lot of the time you'll be able to simplify and keep the numbers smaller!


3y/3 = (21)(4) /3

See how the 3 cross cancels into the 21?
so y = 28


ALWAYS SIMPLIFY THE FRACTIONS FIRST!

WORD PROBLEMS WITH PROPORTIONS:


It's all about setting up the LABELS first!


label A _____________ = ____________ label A

label B xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxlabel B

Math 6H ( Period 3, 6, & 7)

Fractions 6-1

The symbol 1/4 can mean several things:
1) It means one divided by four
2) It represents one out of four equal parts
3) It is a number that has a position on a number line.



1/8 means 1 divided by 8 or 1 ÷ 8
A fraction consists of two numbers
The denominator tells the number of equal parts into which the whole has been divided.
The numerator tells how many of these parts are being considered.
we noted that we could abbreviate ...

denominator as denom with a line above it

and numerator as numer

we found that you could add

1/3 + 1/3 + 1/3 = 3/3 = 1
or 1/4 + 1/4 + 1/4 + 1/4 = 4/4 = 1
we also noted that 8 X 1/8 = 8/8 = 1

We also noticed that 2/7 X 3 = 6/7


So we discussed the properties
For any whole numbers a, b,and c with b not equal to zero

1/b + 1/b + 1/b ... + 1/b = b/b = 1 for b numbers added together

and we noticed that b X 1/b = b/b = 1
we also noticed that
(a/b) X c = ac/b

We talked about the parking lot problem on Page 180

A count of cars and trucks was taken at a parking lot on several different days. For each count, give the fraction of the total vehicles represented by
(a) cars

(b) trucks

Given: 8 cars and 7 trucks
We noticed that you needed to find the total vehicles or 8 + 7 = 15 vehicles
so

(a) fraction represented by cars is 8/15
(b) fraction represented by trucks is 7/15


What if the given was: 15 trucks and 32 vehicles
This time we need to find how many were cars. so 32 -15 = 17 so 17 cars

(a) fraction represented by cars is 17/32
(b) fraction represented by trucks is 15/32

Pre Algebra ( Period 1)

RATIOS AND RATES: 6-1



Ratios = fractions with meaning (it's all about the labels!)
3 ways to write a ratio:


EXAMPLE: 16 girls and 14 boys at a party


16 girls to 14 boys or

16 girls: 14 boys or
16 girls/14 boys


You can simplify this just like a fraction:
8 girls to 7 boys


In fact, anything you can do with a fraction, you can do with a ratio!



Rates = ratios with 2 DIFFERENT LABELS

Miles per gallon, miles per hour

10 out of 16 girls went to my party is not a rate.
Not 2 different labels! (but it is a ratio)



Unit rates = rates with a denominator of 1
(SO MUST HAVE 2 DIFFERENT LABELS)


I drive 150 miles in 3 hours is a rate

To change it to a UNIT RATE, simply DIVIDE the numerator by the denominator
150 miles/3 hours = 50 miles per hour

NOW IT'S A UNIT RATE


People focus on MPGs these days when they buy cars!

A Honda Civic = 40 mpg while a Hummer = 8 mpg




A special unit rate called the UNIT PRICE:
I USE UNIT RATES ALL THE TIME WHEN I TRY TO DECIDE WHETHER IT'S WORTH GOING TO COSTCO INSTEAD OF PAVILIONS
Goldfish = $7.99 at Pavilions for 33.5 oz and $10.99 at Costco for 48 oz.



If you divide $/oz you get a unit rate know as UNIT PRICE
MONEY MUST BE THE NUMERATOR!!!!






IN CLASS: Chapter 6-2: Proportions
A proportion = 2 equal ratios (2 equivalent fractions)

Solve using equivalent fractions or


Cross multiplication and then a one-step equation
(see if you can simplify the fractions before multiplying)

Example: Solve the proportion for y:


4/3 = y
/21

EQUIVALENT FRACTION APPROACH:


Multiply both top and bottom by 7, y = 28



CROSS PRODUCTS APPROACH:
You'll get 3y = (21)(4)
Now divide each side by 3.

Do this before multiplying on the right side!

Why? Because a lot of the time you'll be able to simplify and keep the numbers smaller!


3y/3 = (21)(4) /3

See how the 3 cross cancels into the 21?
so y = 28


ALWAYS SIMPLIFY THE FRACTIONS FIRST!

Algebra Period 4

SLOPE
: 7-4

First, let's talk about what the word "slope" means in the real world:


You can think of the slope of a line as the slope of a ski mountain -


When you're climbing up, it's positive


When you're sliding down, it's negative
 (if you're looking at the mountain from left to right)


The steeper the mountain, the higher the slope value


(A slope of 6 would be an expert slope because it
is much steeper than a slope of 2 which would be an intermediate's slope)


"Bunny slopes" for beginners will be lower numbers,
 generally fractional slopes (like 1/2 or 2/3)



A good benchmark to know is a slope of 1 or -1 is a 45 degree angle


You can also think of slope as rise/run - read this "rise over run"


Rise is how tall the mountain is (the y value)


Run is how wide the mountain is (the x value)


A 1000 foot high mountain (the rise) is very steep if it's only 200 feet wide (the run) (slope = 5)


Another mountain that is also 1000 feet high is not very steep if it is 2000 feet wide (slope = 1/2)


You can think of slope as a calculation using 2 coordinates.

We use m to denote slope:


m = rise/run

change in y
change in x

Rise = Change in y value = Difference in y value = y₂ - y₁
Run = Change in x value = Difference in x value = x₂ - x₁
THE LITTLE 1 AND 2 SUBSCRIPTS JUST STAND FOR Y VALUE OF THE FIRST POINT AND Y VALUE OF THE 2ND POINT.

To calculate slope you need 2 coordinates.

It doesn't matter which one you start with.

Just be consistent!

If you start with the y value of the FIRST point, make sure you start with the x value of the FIRST point!

If you start with the y value of the SECOND point, make sure you start with the x value of the SECOND point!



You can count the slope of a line:


1) Beginning with one point, count up to another point; however far that is, make that the numerator of your slope (because the y value of slope is the numerator)


2) Now count how far over the point is across - You'll need to either go right or left.


Make this the denominator of your slope (because the x value is the denominator of slope)


If you went to the RIGHT, the value is POSITIVE (x values going to the right or positive)


If you went to the LEFT, the value is NEGATIVE



Special slopes:


Horizontal lines in the form of y = have slopes of zero (they're flat!)


Vertical lines in the form of x = have no slope or undefined because the denominator is zero

Math 6H ( Periods 3, 6, & 7)

Graphing Inequalities

You will need to look at the graphs in your textbook. .. page 397

Whenever we graph relations that are inequalities we must be aware of all the facts that can influence your work. You need to ask yourself, "What kind of numbers is the solution supposed to be?"
When you graphed inequalities such as
-3 < x < 2 where x was an integer we used a point on the number line for each integer that could be a solution to that inequality.

To show every number in x < 2
we would use a number line and place an Open Dot at 2 indicating that 2 was NOT part of the solution and then draw a darkened ray away from 2 indicating 1, 0, -1, -2... were all part of the solution.
To show that this line has infinite solutions in that direction, you MUST place an arrow at the end of that darkened ray.

If the inequality was a " less than or equal to" " ≤" you would use a Closed Dot at 2 to indicate that 2 was part of the solution.

We can now graph inequalities such as y ≥ x + 2
first you find the BOUNDARY LINE which is just y = x + 2 and you can use the 3 column table as we have done before or use a T chart as shown in class.

Remember you only need 2 points to determine a line---> but 3 points will help you make sure you have 3 correct points on the line!!

I am going to try to set up a T chart using "I" to separate the x and y
X I Y
-2 I 0
-1 I 1
0 I 2
1 I 3
2 I 4

(Note: it doesn't line up well here.. but hopefully you get the idea)

Plot those points on the graph and you have what appears to be a straight line. Since we are graphing y ≥ x + 2 we ARE including the line so we draw a solid line.

But.. what points are included?
Well, we know that (-2,0) works but we also see if we plug into our inequality that (-2,1) and (-2,2) work as well.

We need to shade the part above the line to indicate all those points are part of the solution as well.

Three set method for graphing an inequality

(1) Determine the boundary line. Draw it--
use a solid line if the boundary line is part of the graph (≤ or ≥)
use a dashed line if the boundary line is NOT part of the graph (< or >)

(2) Shaded either the part above the boundary line or the part below the boundary line.
If the inequality reads y > or y ≥ shade ABOVE the line.
If the inequality reads y < or y ≤ shade BELOW the line

(3) Always CHECK- choose a point you think works within the shaded region and see if it does work.. or use (0,0) and determine if it is part of the solution or not!!

Algebra Period 4

GRAPHING LINEAR EQUATIONS: 7-2

How do you determine whether a given number is a solution?

Plug it in, plug it in, plug it in!



How do you find a solution to an equation yourself?


Plug in for x and find y!


You can use ANY number for x


Then plug in your number and find y



How can you graph a linear equation?


Make an x/y table of values and then graph the coordinates.


You only need 3 coordinates to make a good line!


(The 3rd coordinate serves as a "check" for the other two...in case you made a mistake!)


I always try x = zero and y = zero first because it's usually easy.

Then pick another easy x value!


If this doesn't work well (you get a fraction as an answer and that's not easy to graph),
then try setting x equal to 1, then 2, then 3



LINEAR EQUATIONS: 7-3

What do they look like (what is not a linear equation?)


The variable is to the 1 power -> like x, or y, or a, or b


What is not a linear equation? the variable is not to the 1 power - like x2, x3, etc, or 1/x (x-1)



Two ways to graph:


1) 3 points using a table (like Ch 7-2)


EXAMPLE: 2x - 3y = -6


x y


0 2


3 4


-3 0




2) 2 points using the y and x intercepts (where the line intersects the y and x axis)


Standard form of a linear equation:

Ax +By = C
 where A, B and C should not be fractions


A should be positive (y will be positive or negative)


We won't be using this form to look at the slope of the line!


This is a good format for finding the x and y intercepts!



If it's in standard form, this way works great if both the x and y coefficients are factors of the constant on the other side of the equal sign.



EXAMPLE: 2x - 3y = -6


If x = 0, y = 2


If y = 0, x = -3



Special linear equations:


Ones that are parallel to either the x or the y axis:


Lines parallel to the y axis are vertical lines:


They end up as the form x = with no y variable in the equation at all!


EXAMPLE: x = 4 ends up as a vertical line at x = 4


Still don't get this???


Pick of few points with the x value of 4:
(4, 0) (4, 2) (4, -3)


Graph those and join them in a line.
 What do you get???


A vertical line!




Lines parallel to the x axis are horizontal lines:


They end up as the form y = with no x variable in the equation at all!


EXAMPLE: y = 4 ends up as a horizontal line at y = 4


Still don't get this???


Pick of few points with the y value of 4:
(0, 4) (2, 4) (-3, 4)


Graph those and join them in a line.
What do you get???


A horizontal line!

Math 6H ( Periods 3, 6, & 7)

Graphs of Equations 11-9

An equation in two variables y = x + 2
produces an infinite number of ordered pairs
If we give x the value of 3, a corresponding value of y is determined
y = (3) + 2 = 5
The ordered pair is (3, 5)
If we let x = 4
y = (4) + 2 = 6
and we get the ordered pair (4, 6)
What happens if x = 0
y = (0) + 2 = 2 ( 0, 2)
or x = -2
y = (-2) + 2 = 0 ( -2, 0)
For each value of x there is EXACTLY 1 value of y.
set of ordered pairs in which no two ordered pairs have the same x is called a FUNCTION
I like to remember ordered pairs---> ( ordered, pairs)

y = 2x -3
in the future you will see it written as
f(x) = 2x -3
so if x = 3
f(3) = 2(3) -3 = 6-3 = 3
so f(3) = 3
if x = 5
f(5)= 2(5) - 3 = 10 -3 = 7
so f(5) = 7

We used a three column chart to compute our ordered pairs.
Please refer to the blue sheet glued into your spiral notebook for the examples we completed in class-- if you were absent, please come in one morning and I will review that chart with you.


The following equations create curves that are called PARABOLAS!! Notice the difference in these equations from our previous equations
y = x² +1
when we create your three column table using integers from -2 to 2
we notice
y = (-2)² +1 = 4 + 1 = 5 ordered pair (-2, 5)
y = (-1)² +1 = 1 + 1 = 2 ordered pair (-1, 2)
y = (0)² +1 = 0 + 1 = 1 ordered pair (0, 1)
y = (1)² +1 = 1 + 1 = 2 ordered pair (1, 2)
y = (2)² +1 = 4 + 1 = 5 ordered pair (-2, 5)

When you graph this... you get a "U" shaped graph.

Remember linear equations LINEar equations are lines!1
and look like y = x + 2

PARABOLAS have the form y = x² or y = -x²

Let's try
y = 2 -x²
With our 3 column table
for values of x from -2 to 2
we find
y = 2 -(-2)² = 2 -(4) = -2 and the ordered pair is (-2,-2)
y = 2 -(-1)² = 2 - (1) = 1 and the ordered pair is ( -1, 1)
y = 2 -(0)² = 2 - 0 = 2 and the ordered pair is (0, 2)
y = 2 -(1)² = 2 -1 = 1 and the ordered pair is (1, 1)
y = 2 -(2)² = 2 - (4) = -2 and the ordered pair is (2, -2)

When you graph these ordered points you find you have an upside down U
hmmm... y = -x² results in a sad face parabola
and y = x² results in a happy face parabola!!

Algebra Period 4

GRAPHING ORDERED PAIRS:
7-1

Review of x y Coordinate Plane Graphing from Pre-Algebra

Cartesian plane: Named after French mathematician Descartes.


plane: a two dimensional (across and up/down) flat surface that extends infinitely in all directions.


quadrant: 2 perpendicular lines called axes split the plane into 4 regions....

quad means 4
quadrant

names: begin in the top right (where you normally write your name!) and go counterclockwise in a big "C" (remember it for "C"oordinate)


They are named I, II, III, IV in Roman Numerals



coordinate - A coordinate is the position of a point in the Cartesian plane


coordinate = "co" means goes along with (COefficient, COworker, CO-president, CO-champions)



"ordinate" means in order

So coordinate means numbers that go along with each other in a certain order


The numbers are the x and y values and the order is that the x always comes first



Also called an ordered pair (x y "ordered" and they are a "pair" of numbers)


Ordered pairs are recognized by the use of ( x , y ) format



origin = (0, 0) the center of the graph (its beginning or origin)


When you count the coordinate' s position, you count from the origin.



x comes before y in the alphabet so the order is (x, y) 
always go right or left first, then up or down



the x axis is the horizontal axis (goes across)


Remember that because the number line also is horizontal and you learn that first


(the pattern to remember is x is always first and the number line is before going up and down)



NOW LET'S GET TO WHAT YOU ACTUALLY DO!!!


1) Count your x value:


positive x, count right from origin (positive numbers are to the right of zero on number line)


negative x, count left from origin


2) Count your y value:


positive y value, count up from where your x value was (up is the positive direction)


negative y value, count down from where your x value was (down is the negative direction)



EXAMPLE:
(3, 5) Count 3 to the right from the origin, then 5 up


(3, -5) Still count 2 to the right, but now count 5 down


(-3, 5) Count 3 to the left from the origin, then count 5 up


(-3, -5) Again count 3 to the left, but now count 5 down



BUT WHAT HAPPENS WHEN
 ONE OF THE VALUES IS ZERO?


If the y value is zero it means that you move right or left, but don't go up or down:


SO YOUR POINT WILL BE ON THE x AXIS........

x axis is where y = 0 
Example: (3, 0) is a point on the x axis, 3 places to the RIGHT


Example: (-3, 0) is a point on the x axis, 3 places to the LEFT



If the x value is zero it means that you don't move right or left, you just go up or down.


SO YOUR POINT WILL BE ON THE y AXIS...........

y axis is where x = 0 


Example: (0, 3) is a point on the y axis, 3 places UP


Example: (0, -3) is a point on the y axis, 3 places DOWN