Math 6A ( Periods 2 & 5)

Writing Expressions 3.2 

 Math Operations  vocab of what means addition, subtraction, multiplication, and division
Make sure to glue the '1/2 sheet' of math word phrases that we associate with each of the four basic operations -- into your spiral notebook (SN)

Use a "LET STATEMENT"
Let n = the number
That is the simplest "LET STATEMENT.  You are defining your variable!


We can use the same mathematical expression to translate many different word phrases
Five less than a number n
The number n decreased by five
The difference when five is subtracted from a number n

All three of those phrases can be translated into the variable expression

n - 5

The product of a number and 3   3n
4 times a number  4n
The quotient of 3 and a number  3/n
The quotient of a number y divided by ten becomes y/10. It may look like only a fraction to you-- but if you read y/10 as always " y divided by 10" you have used the proper math language.

Twelve more than three times a number m

Wait-- where are you starting from... in this case you are adding 12 to 3m so you must write

3m + 12


Not all word phrases translate directly into mathematical expressions. Sometimes we need to interpret a situation.. we might need to use relationships between to help create our word phrase.
10 less than a number is n-10
a number increased by 9  n + 9
9 more than a number is also n + 9



In writing a variable expression for the number of hours in w workdays, if each workday consists of 8 hours...

8w would be our expression


Some everyday words we use to so relationships with numbers:
consecutive whole numbers are whole numbers that increase by 1 for example 4, 5, 6
So the next consecutive whole number after w is w + 1.

A preceding whole number is the whole number that is 1 less and the next whole number is the whole number that is 1 greater.

So the number which precedes x would be x - 1.
The next number after n is n + 1



What if I asked what is the next consecutive EVEN number after the even number "m"
It would be m + 2



What would it be if I asked what was the next consecutive odd number, after the odd number x?
x + 2


What if I asked what is the next consecutive EVEN number after the even number "m"
It would be m + 2


What would it be if I asked what was the next consecutive odd number, after the odd number x?
x + 2
Let's look at those relationships like the workdays from Monday's lesson... We are going to find another strategy to use for some of the more complicated expressions...

First set up a T chart- as discussed in class
put the unknown on the left side of the T chart... The unknown is always the one that reads like " w workdays"

so in this case

w workdays on the left side and under it you put
1
2
3

On the right side put the other variable-- in this case hours
under hours put the corresponding facts you know-- the relationship between workdays and hours as given in this case
hours
8
16
24

all of those would be on the right side of the T chart.

Now look at the relationships and ask yourself--
"What do you do to the left side to get the right side?"

and in this case, specifically...

What do you do to 1 to get 8?
What do you do to 2 to get 16?
What do you do to 3 to get 24?

Do you see the pattern?

For each of those the answer is "Multiply by 8" so
what do you do to w-- The answer is Multiply by 8
so the mathematical expression in this case is "8w."

What about writing an expression for how many dimes are in x dollars?

Use your T- chart
x dollars    dimes
1..................10
2..................20
3..................30
x...................10x

Put 3 known relationships and then ask yourself "What did I do to the left side to get the right side"
What did I do to 1 to get 10? ... answer multiply by 10
What did I do to 2 to get 20? ... answer multiply by 10
What did I do to 3 to get 30?  ... answer multiply by 10
So What must I do to x? it must be the same  ... answer multiply by 10  so 10x is what I am looking for!

How many dollars do we have in n nickles?
set up-- unknown on the left!
n nickles      dollars
20....................1
40....................2
60....................3
n......................?
Again, Ask yourself  "What did I do to the left side to get the right side"
What did I do to 20 to get 1? ... divide by 20
What did I do to 40 to get 2?  ... divide by 20
What did I do to 60 to get 3?  ... divide by 20
So what must I do to n? it must be the same  ... divide by 20 so n/20

The number of feet in i inches

i inches is the unknown... so that goes on the left side of the T chart... with feet on the right

i inches ___feet
12...............1
24...............2
36...............3

I filled in three known relationships between inches and feet Now, ask your self those questions again...
"What do you do to the left side to get the right side?"

and in this case, specifically...

What do you do to 12 to get 1?
What do you do to 24 to get 2?
What do you do to 36 to get 3?

In each of these, the answer is divide by 12
so What do you do to i? the answer is divide by 12
i inches ___feet
12...............1
24...............2
36...............3
i................i/12



and it is written i/12

Math 6A ( Periods 2 & 5)

Algebraic Expressions 3.1 

An algebraic expression is an expression that may contain numbers, operations and one or more symbols (called variables) A variable is a symbol used to represent one or more numbers. The numbers are called the values of the variable.


Parts of an algebraic expression are called TERMS
5p + 4


When we write a product that involves a variable, we usually omit the multiplication symbol (whether that be written as x or as ∙ or even with parentheses). Thus, 3 x n is written as 3n
and 2 x a x b is written as 2ab

In numerical expressions for products a multiplication symbol must be used to avoid confusion.

9 x 7 may be written as 9 ∙ 7 or even 9(7)

5p + 4
Let's look at the term 5p
The number that modifies the variable is called a coefficient  Our book defines it as the numerical factor of a term that contains a variable
now let's look at the term +4 It is a term without a variable and it is called a constant.
We will
a)identify terms,
b)name the coefficients ,
c) name the constants

5x + 13                     a)5x , +13                b) 5              c) 13
2x²  + y + 3              a) 2x²  ,y ,  3            b) 2, 1           c) 3
4x²  -3y + z - 5      a) 4x²  ,-3y, z,  - 5     b) 4, -3, 1        c) -5


Using Exponents
d∙d∙d∙d = d⁴ 
1.5h∙h∙h = 1.5h³

h∙h∙h∙o∙o∙o= h³o³ = (ho)³ = hohoho!

When a mathematical sentence uses an equal sign, it is called an equation. An equation tells us that two expressions name the same number. The expression to the left of the equals sign is called the left side of the equation and the expression to the right of the equals sign is called the right side.
expression = expression

When a number is substituted for a variable in the variable expression and the indicated operation is carried out, we say that the variable expression has been evaluated. For example, if n has the value 6 in the variable expression 3n, then 3n has the value 3 (6), or 18
Example: Evaluate the expression 6a when the variable has the following values:
6a; 2, 4, 6, 8
You would substitute in each value for the variable a
6(2) = 12
6(4) = 24
6(6) = 36
6(8) = 48

148 ÷ 4 =
148/4
37


if m = 3 and n = 18
n ÷ m
substitute in
n/m or (18)/(3) = 6


If y = 18 and x = 8
4y ÷ 3x immediately set this up as
4y/3x

Now substitute in your values
4(18) / 3(8)
72/24 = 3

Evaluate k + 10 when k = 25
(25) + 10 = 35
When k = -25
(-25) + 10 = -15
Evaluate 4n when n = -12
4(-12) = -48
Now  a÷b   when a= 16 and b = 2/3
Yikes... think
16 ÷ 2/3  WOW-- instead of dividing we think multiply by its reciprocal.
The reciprocal of 2/3 is 3/2
so 16 ( 3/2) = 24
What happens with
3x-14 if x = 5
3(5) - 14 = 1
You must use the ORDER OF OPERATIONS
z² + 8.5 when z = 2
(2)² + 8.5
4 + 8.5
12.5
What about 30 - 24÷  y  when y = 6
THINK:  Order of Operations
30 - 24 ÷ 6
30 - 4 = 26
Saving for a Skateboard
It's on sale for $125. You have $45.00 and you are saving $3.00 each week.
How could we write an expression for how much you could save after w weeks?
We made a chart for weeks 1-4
and then realized we could write
45 + 3w
Why did we put the 3 in front of the w?
The three represents the $3 we save each week-- and you always put the coefficient first
How much did you save after
4 weeks? Substitute 4 for w--> 45 + 3(4) = 57  $57
10 weeks? 45 + 3(10) = 75  $75
20 weeks? 45 + 3(20) = 105  $ 105
What would be the expression if you could save $5 a week?
45 + 5w

Algebra ( Periods 1 & 4)

Chapter 4-3 Point Slope Form 

There is another way to solve for a line other than using slope-intercept form  Although most people use the slope- intercept form for all cases, the POINT-SLOPE Form  is actually easier—everything is built into the format!  You do not need to “put it altogether” at the end!

MY FAVORITE!!

It gives you exactly a point and the slope—just by looking at it!

You do not need to do anything BUT graph!!

Point- Slope Form of a line:

y-y₁ = m ( x- x₁)

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

y – 1 = 2( x – 3)

What you have now is point-slope form of the line!

You can graph easily starting at (3, 1) and going up 2 and to the right 1

You can also simplify this and get the slope- intercept form of the line:

y - 1 = 2x – 6

y = 2x – 5

If you are trying to link the Slope-Intercept form to the Point-Slope form of the line:  The point slope version eliminates one step from using the slope intercept form

In the slope intercept form, you plug in the point and the slope and solve for b—
AND THEN rewrite the equation using the intercept that you found.

In point slope form, once you plug in the point and slope, you just simplify and the equation is already done!

IF you don’t have the slope, you will need to compute it with the formula—just like you did with Slope Intercept FORM

The biggest difference is that any point is plugged into this form, while  the Slope- Intercept Form focuses on the ONE specific point on the y axis 

Algebra Honors (Period 6)

Chapter 4-2 Writing Equations in Slope-Intercept Form 

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

We know that y = mx + b so we can substitute that in what we know (what the problem gives us as information) and solve for whatever we are missing

It helps to memorize this little rhyme (Mrs Sobieraj made it up!)

Oh mystery line,
What could you be?
If I could just find you,
 y = mx + b
First I’ll find m,
Then I’ll find b
Then I’ll put it all together
And I will 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—(carefully pick two sets of integer points)

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) Remember: Put it all together in ONE equation at the end!

There are FIVE general cases of mystery lines

First Case:
You are given the slope and the y intercept (that is the easiest case)
For example:  you are given 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 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 the FOURTH CASE (below) completely!

Put the information together in y = mx + b form

Third Case:
You are given a point and the slope and need to find the intercept ( b)
Example: ( 3, 1)is a point on the line and m = 2

Plug in the point and the slope and find b

That is, start with y = mx + b
You have a point ( 3, 1) plug it in to that equation:

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

-5 = b or
b = -5

Now put it altogether with the given slope of m = 2 and the y intercept ( b) which you just found

y = 2x – 5

Fourth Case:
You are given a point and the y intercept and need to find the slope > Let’s use the point ( 3, 1) again but this time you are given b = 2

Again you can use y = mx + b . This time, however you are solving for m ( the slope)
1 = 3m + 2
-1 = 3m

-1/3 = m
m = -1/3

Again, NOW put it all together with the given intercept and the slope you just found
 y = (-1/3)x + 2

Fifth Case:
You are given 2 points on a line and need to find the slope and the y intercept
Example: ( 1, 3) and ( -2, -3) are 2 points on the line

You first need to find the slope using the formula
m = change in y/ change in x

m = (-3 -3)/(-2-1)  or  (3--3)/(1--2)  which really is (3+3)/(1+2) or
6/3 = 2

Now plug the slope in with one ( you get to pick—it will work with either) 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

Chapter 4-3 Point Slope Form 

There is another way to solve for a line other than using slope-intercept form  Although most people use the slope- intercept form for all cases, the POINT-SLOPE Form  is actually easier—everything is built into the format!  You do not need to “put it altogether” at the end!

MY FAVORITE!!

It gives you exactly a point and the slope—just by looking at it!

You do not need to do anything BUT graph!!

Point- Slope Form of a line:

y-y₁ = m ( x- x₁)

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

y – 1 = 2( x – 3)

What you have now is point-slope form of the line!

You can graph easily starting at (3, 1) and going up 2 and to the right 1

You can also simplify this and get the slope- intercept form of the line:

y - 1 = 2x – 6

y = 2x – 5

If you are trying to link the Slope-Intercept form to the Point-Slope form of the line:  The point slope version eliminates one step from using the slope intercept form

In the slope intercept form, you plug in the point and the slope and solve for b—
AND THEN rewrite the equation using the intercept that you found.

In point slope form, once you plug in the point and slope, you just simplify and the equation is already done!

IF you don’t have the slope, you will need to compute it with the formula—just like you did with Slope Intercept FORM

The biggest difference is that any point is plugged into this form, while  the Slope- Intercept Form focuses on the ONE specific point on the y axis 

Algebra ( Periods 1 & 4)

Chapter 4-2 Writing Equations in Slope-Intercept Form 

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

We know that y = mx + b so we can substitute that in what we know (what the problem gives us as information) and solve for whatever we are missing

It helps to memorize this little rhyme (Mrs Sobieraj made it up!)

Oh mystery line,
What could you be?
If I could just find you,
 y = mx + b
First I’ll find m,
Then I’ll find b
Then I’ll put it all together
And I will 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—(carefully pick two sets of integer points)

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) Remember: Put it all together in ONE equation at the end!

There are FIVE general cases of mystery lines

First Case:
You are given the slope and the y intercept (that is the easiest case)
For example:  you are given 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 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 the FOURTH CASE (below) completely!

Put the information together in y = mx + b form

Third Case:
You are given a point and the slope and need to find the intercept ( b)
Example: ( 3, 1)is a point on the line and m = 2

Plug in the point and the slope and find b

That is, start with y = mx + b
You have a point ( 3, 1) plug it in to that equation:

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

-5 = b or
b = -5

Now put it altogether with the given slope of m = 2 and the y intercept ( b) which you just found

y = 2x – 5

Fourth Case:
You are given a point and the y intercept and need to find the slope > Let’s use the point ( 3, 1) again but this time you are given b = 2

Again you can use y = mx + b . This time, however you are solving for m ( the slope)
1 = 3m + 2
-1 = 3m

-1/3 = m
m = -1/3

Again, NOW put it all together with the given intercept and the slope you just found
 y = (-1/3)x + 2

Fifth Case:
You are given 2 points on a line and need to find the slope and the y intercept
Example: ( 1, 3) and ( -2, -3) are 2 points on the line

You first need to find the slope using the formula
m = change in y/ change in x

m = (-3 -3)/(-2-1)  or  (3--3)/(1--2)  which really is (3+3)/(1+2) or
6/3 = 2

Now plug the slope in with one ( you get to pick—it will work with either) 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

Algebra Honors ( period 6)

Chapter 3 -5 Arithmetic Sequences as Linear Functions

An arithmetic sequence is an ordered list of numbers ( called terms) where there is a common difference (d) between consecutive terms. the common difference can either be positive (increasing)  or negative ( decreasing)

Because an arithmetic sequence has a constant difference, it is a linear function. There is a formula using this common difference to find the equation of any arithmetic sequence:

a_n = a₁ + ( n-1)d

Notice that it is saying that any term in the sequence, a_n,  can be found by adding 1 less than the number of terms of the common difference to the first term, a_1.   Why 1 less than the number of the terms you want?

Take the following arithmetic sequence:

…17, 21, 25,29, 33….

If it’s the 2^nd term, there is only 1 common difference of 4 between the 2 terms. If it is the 3^rd term, there would be 2 common differences of 4 between the 3  terms, etc.

n is always positive because it represents the number of terms and that can’t be negative.

Where  a_n  represents any term in the sequence and a₁ represents the first term in a sequence, n represents the number of the terms in a sequence and d represents the common difference between consecutive terms in a sequence.

This is called an EXPLICIT FORMULA You can explicitly find any term number in the sequence as long as you know the 1^st term and the common difference. For example if the the common difference is 4 and the first term is 1 and you are trying to find the 27^th term

a_n = a₁ + ( n-1)d

a₂₇ = 1 + ( 27-1)4

a₂₇ = 1 + ( 26)4

a₂₇ = 1 + ( 104

a₂₇ = 105

You can also find the next term in the sequence if you know the
RECURSIVE FORUMULA

This is a Function Rule that tells you what the relationship between consecutive terms is. For example if the common difference is 4 and your last term is 101 the next term is 105—without knowing any of the other preceding terms. The terms in the sequence are shown as a list with 3 periods (called an ellipsis) at the end showing that it continues infinitely.

For example, the arithmetic sequences for the above example was

1, 5,9, 13, …

Graphing the terms of an arithmetic sequence shows that it is a linear function

a_n = 1 + ( n-1)4

Simplify

a_n = 1 + ( n-1)4

a_n = 1 + 4n -4

a_n = 4n -3

Now just substitute y for a_n   and x for n

y = 4x – 3

Notice that d is now the slope and also notice that the domain of the sequence is the natural numbers (the counting numbers) because you can’t have a negative term number!

Graph 3 points using 1, 2, 3 for the first 3 terms

(1, 1) ( 2,5) (3, 9)

Algebra ( Periods 1 & 4)

Chapter 3-4 Direct Variation

We’ve learned that the unit rate is the constant rate of change in a linear relationship and that it’s the slope of a line when it’s graphed. We’ve also learned that if a graph of an equation goes through the origin (0,0)  it’s proportional  and the ratio of any y value to it’s x value is a constant (which turns out to be the unit rate or constant rate of change or slope of the line)

When the linear relationship is proportional, we say it’s a DIRECT VARIATION. Now the constant rate of change, the slope, the unit rate, is called the CONSTANT OF VARIATION or the CONSTANT OF PROPORTIONALITY

This is not a new concept. IT IS  just NEW VOCAB!

We also say: y varies directly (constantly) with x.

The slope is now replaced by the letter k instead of m

Finding the equation of a line that is proportional

Find k (the slope) by counting the rise/run of the graph

Write the equation using the format  y = kx

Notice: if you always pick the origin as the point to count rise/run from—the slope (k) is always just y/x

In a word problem, if it says one amount VARIES DIRECTLY with another, you know that the origin is one of the points!!

You also know that the equation is y = kx
YOU just need to find k
and k is y/x of any point OTHER THAN THE ORIGIN

A babysitting example

The amount of money earned  VARIES DIRECTLY with the time worked.

THINK: the graph and equation go through (0,0)

THINK: Any other point will give you the slope, or constant of proportionality, or unit rate ( all the same thing) SO you only need one additional point.

We are given that she earns $30 for 4 hours. Find the equation.

Rise/Run = y/x
BECAUSE THEY SAID IT VARIED DIRECTLY!!

k = 30/4
Simplify
k = 7.5
So the equation is y = 7.5x

What does the 7.5 represent?
The unit rate of $7.50/ hour of babysitting!

A bicycling example 
The distance the cyclist bikes in miles VARIES DIRECTLY with the time in hours that he bikes.

THINK: The graph and equation go through the origin (0,0).THINK: Any other point will give you the slope, or constant of proportionality, or unit rate (all the same thing) SO you only need one additional point.

He bikes 3 miles in ¼ hour. Find the equation.

Rise/run = y/x
BECAUSE THEY SAID IT VARIES DIRECTLY

k = 3/¼  or 3/.25 Now the hardest part is doing this 3/.25

If you kept it as 3/¼  you could read this as 3 divided by ¼
THINK: instead of dividing, multiply by the reciprocal of ¼

or 3 (4/1) = 12 (Wait, wasn’t that much easier than dividing 3 by .25!!
k = 12

The equation is y = 12x

What does the 12 represent?

The unit rate of 12 miles/ hour – that’s the cyclist’s speed 12mph

  Determining whether a Table of Values is Direct Variation If you are given a table of values, you can determine if the relationship is direct variation by dividing 3 y’s by their x values and making sure that you get the SAME value. If you do, it is proportional, goes through the origin (0,0) and the slope of y/x is the unit rate ( which is now called the constant of variation)!

Example

Given 3 points (5, 20) , (6, 24), and (7, 28):

Divide each y/x

20/5 = 4

24/6 = 4

28/7 = 4

Since all the ratios simplify to the same value (4), it is a direct variation. The slope of 4 is the unit rate, which is the constant rate of change and is now also called the constant of variation.

Finding Additional Values for the Direct Variation once you have the EquationOnce you have the equation y = kx, you can find infinite additional values (points) that will work.
For example, in the first babysitting example, the equation is y = $7.50x, which we write as y = 7.5x  If she babysits for 20 hours, how much did she earn?
x = 20

so y = 7.5(20) = 150 so She earns $150.
If she earns $750, how many hours did she need to work?

Now y = 750  so  750 = 7.5x
It is a one-step equation and we get
x = 100 or 100 hours!

 Finding the Equation if you know 1 point and then Finding Additional Values

y varies directly with x. Write an equation for the direct variation. Then find each value

If y = 8 when x = 3, find y when x = 45

FIRST you need to find k

y = kx… In this case we have 8 = k(3) or 8 = 3k

Solve this 1 step equation—leaving it in fraction form!

8/3= k

so

y = (8/3)x

Now, find y when x = 45

y = (8/3)(45)

solve

y = 120

Applying direct variation to the Distance Formula d = rt

A jet’s distance varies directly as the hours it flies

If it traveled 3420 miles in 6 hours, how long will it take to fly 6500 miles?

k = 3420/6 = 570mph ( its speed)

6500 = 570t

t ≈11.4

about 11.4 hours

Algebra Honors ( Period 6)

Chapter 3-4 Direct Variation

We’ve learned that the unit rate is the constant rate of change in a linear relationship and that it’s the slope of a line when it’s graphed. We’ve also learned that if a graph of an equation goes through the origin (0,0)  it’s proportional  and the ratio of any y value to it’s x value is a constant (which turns out to be the unit rate or constant rate of change or slope of the line)

When the linear relationship is proportional, we say it’s a DIRECT VARIATION. Now the constant rate of change, the slope, the unit rate, is called the CONSTANT OF VARIATION or the CONSTANT OF PROPORTIONALITY

This is not a new concept. IT IS  just NEW VOCAB!

We also say: y varies directly (constantly) with x.

The slope is now replaced by the letter k instead of m

Finding the equation of a line that is proportional

Find k (the slope) by counting the rise/run of the graph

Write the equation using the format  y = kx

Notice: if you always pick the origin as the point to count rise/run from—the slope (k) is always just y/x

In a word problem, if it says one amount VARIES DIRECTLY with another, you know that the origin is one of the points!!

You also know that the equation is y = kx
YOU just need to find k
and k is y/x of any point OTHER THAN THE ORIGIN

A babysitting example

The amount of money earned  VARIES DIRECTLY with the time worked.

THINK: the graph and equation go through (0,0)

THINK: Any other point will give you the slope, or constant of proportionality, or unit rate ( all the same thing) SO you only need one additional point.

We are given that she earns $30 for 4 hours. Find the equation.

Rise/Run = y/x
BECAUSE THEY SAID IT VARIED DIRECTLY!!

k = 30/4
Simplify
k = 7.5
So the equation is y = 7.5x

What does the 7.5 represent?
The unit rate of $7.50/ hour of babysitting!

A bicycling example 
The distance the cyclist bikes in miles VARIES DIRECTLY with the time in hours that he bikes.

THINK: The graph and equation go through the origin (0,0).THINK: Any other point will give you the slope, or constant of proportionality, or unit rate (all the same thing) SO you only need one additional point.

He bikes 3 miles in ¼ hour. Find the equation.

Rise/run = y/x
BECAUSE THEY SAID IT VARIES DIRECTLY

k = 3/¼  or 3/.25 Now the hardest part is doing this 3/.25

If you kept it as 3/¼  you could read this as 3 divided by ¼
THINK: instead of dividing, multiply by the reciprocal of ¼

or 3 (4/1) = 12 (Wait, wasn’t that much easier than dividing 3 by .25!!
k = 12

The equation is y = 12x

What does the 12 represent?

The unit rate of 12 miles/ hour – that’s the cyclist’s speed 12mph

  Determining whether a Table of Values is Direct Variation If you are given a table of values, you can determine if the relationship is direct variation by dividing 3 y’s by their x values and making sure that you get the SAME value. If you do, it is proportional, goes through the origin (0,0) and the slope of y/x is the unit rate ( which is now called the constant of variation)!

Example

Given 3 points (5, 20) , (6, 24), and (7, 28):

Divide each y/x

20/5 = 4

24/6 = 4

28/7 = 4

Since all the ratios simplify to the same value (4), it is a direct variation. The slope of 4 is the unit rate, which is the constant rate of change and is now also called the constant of variation.

Finding Additional Values for the Direct Variation once you have the EquationOnce you have the equation y = kx, you can find infinite additional values (points) that will work.
For example, in the first babysitting example, the equation is y = $7.50x, which we write as y = 7.5x  If she babysits for 20 hours, how much did she earn?
x = 20

so y = 7.5(20) = 150 so She earns $150.
If she earns $750, how many hours did she need to work?

Now y = 750  so  750 = 7.5x
It is a one-step equation and we get
x = 100 or 100 hours!

 Finding the Equation if you know 1 point and then Finding Additional Values

y varies directly with x. Write an equation for the direct variation. Then find each value

If y = 8 when x = 3, find y when x = 45

FIRST you need to find k

y = kx… In this case we have 8 = k(3) or 8 = 3k

Solve this 1 step equation—leaving it in fraction form!

8/3= k

so

y = (8/3)x

Now, find y when x = 45

y = (8/3)(45)

solve

y = 120

Applying direct variation to the Distance Formula d = rt

A jet’s distance varies directly as the hours it flies

If it traveled 3420 miles in 6 hours, how long will it take to fly 6500 miles?

k = 3420/6 = 570mph ( its speed)

6500 = 570t

t ≈11.4

about 11.4 hours

Algebra ( Periods 1 & 4)

Chapter 4-1 Graphing Equations in Slope-Intercept Form

The most used form of a linear equation: 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 crosses the y axis)

The m and b are called the PARAMETERS of the equation.

Solve the equation for “y” means isolate the y on one side of the equal signs

Example:  -3y = -2x  - 6

If you tried graphing randomly, (setting up a small table or t chart) you would problem select the x points of 0, 1, 2.  Only when x = 0 will give you an integer value for y. All the other y values are fractions/ decimals à which makes it difficult to graph accurately!
 

The slope- intercept form provides the explanation for this.

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!) YAY!!!

Use the counting method for slope on your graph, you should have counted:
UP  2 and RIGHT 3. The slope therefore is 2/3
Look at the equation—it told you the slope was 2/3 without any work (YAY!!)

Graphing when the line is in Slope- Intercept Form

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

1) Graph the intercept on the y axis (That is the positive or negative constant at the end of your equation. Your HOME BASE)

2) Count the next point by using the slope of x coefficient as a fraction (so if you have an integer—place it “over” 1)

For the equation y = (2/3)(x) + 2

1) graph  a point at (0,2)

2) 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” or “Rise over Run”
The numerator is the change in y  and the denominator is the change in x.

If it is positive you are counting up (positive) and to the right (positive) OR
you can count down (negative) and to the left (negative) because when multiplying 2 negatives become positive

If it is negative you are counting down (negative) and to the right (positive) OR
you can count up (positive) and to the left (negative) because then you would have a positive ∙ negative = negative

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
(Notice; x and y are on the same side of the equal sign, x is positive, there are NO fractions, and the constant is alone)
This equation is NOT easy to graph in this form because your y intercept will not be an integer ( In fact it will not be that easy to graph in slope intercept either—but we will get to that later)

Restate into slope intercept

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 so

m = -3/4 (so you are sliding down at a little less than a 45 degree angle. Remember a slope of 1 or -1 is  45 degrees)

The y intercept is the constant b = 5/2 ( so the line crosses the y axis at 2½)

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 2½ on the y axis and count down 3 and to the right 4. That is actually HARD to get accurate!

Neither 3 nor 4 is a factor of the constant 10 ( 3x + 4y = 10) so the intercepts will also be fractions.

We need the x term to end up with ½ so that when we add that to the b (which is 5/2) we will get an integer.
So try letting x be 2 because that will cross cancel with the -3/4 slope

y = (-3/4)(2) + 5/2 =  -3/2 + 5/2 = 1
so we just found a coordinate that has just integers ( 2, 1)
Graph that point. Now count the slope from THAT POINT ( instead of the y intercept)

HORIZONTAL LINES
You can think of these lines in Slope- Intercept form as y = mx + b IF your REMEMBER that the slope of a horizontal line = 0 àthe equation is y = (0)x + b
We have already learned that the equation of any horizontal line is y = a constant
Except for the horizontal line y = 0 ( which is the x axis) horizontal lines have no x intercept and therefore no intercepts ( no roots, no solutions, no zeros)

Finding the equation of a line by looking at its graph is easy if you can read the y intercept!

Simply plug in the y intercept as  b and then count the rise over run as the slope!

Algebra Honors ( Period 6)

Chapter 4-1 Graphing Equations in Slope-Intercept Form

The most used form of a linear equation: 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 crosses the y axis)

The m and b are called the PARAMETERS of the equation.

Solve the equation for “y” means isolate the y on one side of the equal signs

Example:  -3y = -2x  - 6

If you tried graphing randomly, (setting up a small table or t chart) you would problem select the x points of 0, 1, 2.  Only when x = 0 will give you an integer value for y. All the other y values are fractions/ decimals à which makes it difficult to graph accurately!
 

The slope- intercept form provides the explanation for this.

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!) YAY!!!

Use the counting method for slope on your graph, you should have counted:
UP  2 and RIGHT 3. The slope therefore is 2/3
Look at the equation—it told you the slope was 2/3 without any work (YAY!!)

Graphing when the line is in Slope- Intercept Form

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

1) Graph the intercept on the y axis (That is the positive or negative constant at the end of your equation. Your HOME BASE)

2) Count the next point by using the slope of x coefficient as a fraction (so if you have an integer—place it “over” 1)

For the equation y = (2/3)(x) + 2

1) graph  a point at (0,2)

2) 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” or “Rise over Run”
The numerator is the change in y  and the denominator is the change in x.

If it is positive you are counting up (positive) and to the right (positive) OR
you can count down (negative) and to the left (negative) because when multiplying 2 negatives become positive

If it is negative you are counting down (negative) and to the right (positive) OR
you can count up (positive) and to the left (negative) because then you would have a positive ∙ negative = negative

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
(Notice; x and y are on the same side of the equal sign, x is positive, there are NO fractions, and the constant is alone)
This equation is NOT easy to graph in this form because your y intercept will not be an integer ( In fact it will not be that easy to graph in slope intercept either—but we will get to that later)

Restate into slope intercept

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 so

m = -3/4 (so you are sliding down at a little less than a 45 degree angle. Remember a slope of 1 or -1 is  45 degrees)

The y intercept is the constant b = 5/2 ( so the line crosses the y axis at 2½)

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 2½ on the y axis and count down 3 and to the right 4. That is actually HARD to get accurate!

Neither 3 nor 4 is a factor of the constant 10 ( 3x + 4y = 10) so the intercepts will also be fractions.

We need the x term to end up with ½ so that when we add that to the b (which is 5/2) we will get an integer.
So try letting x be 2 because that will cross cancel with the -3/4 slope

y = (-3/4)(2) + 5/2 =  -3/2 + 5/2 = 1
so we just found a coordinate that has just integers ( 2, 1)
Graph that point. Now count the slope from THAT POINT ( instead of the y intercept)

HORIZONTAL LINES
You can think of these lines in Slope- Intercept form as y = mx + b IF your REMEMBER that the slope of a horizontal line = 0 àthe equation is y = (0)x + b
We have already learned that the equation of any horizontal line is y = a constant
Except for the horizontal line y = 0 ( which is the x axis) horizontal lines have no x intercept and therefore no intercepts ( no roots, no solutions, no zeros)

Finding the equation of a line by looking at its graph is easy if you can read the y intercept!

Simply plug in the y intercept as  b and then count the rise over run as the slope!

Algebra Honors ( Period 6)

Chapter 3-3 Rate of Change and Slope

We’ve already looked at the slope (m) of lines—today we will connect slope to the RATE of the CHANGE of the linear function (the line). the rate of change for a line is a CONSTANT… it is the same value EVERYWHERE on the line

This change, also know as the slope, is found by  finding the rise over the run between ANY 2 points.  rise/run

The rise is the change in y and the run is the change in x.

In a real world example, the rate of change is the UNIT RATE

If you are buying video games that are all the same price on BLACK FRIDAY, two data points might be

# of computer           Total
games                          cost

4                                  $156
6                                  $234

The slope or rate of change  is the  change in y/ the change in x

(234- 156)/ 6-4

78/2

or $39/ video game

Again, as long as the function is linear, or one straight line, it has a constant rate of change, or slope between ANY TWO POINTS

The constant rate of change, or slope, is the rise over the run—or the change in y over the change in x

or
y₂ – y₁/ x₂-x₁

Slope = rise/run ( rise over run)
=change in the y values/ change in the x values =
Difference of the y values/ Difference of the x values

Mrs Sobieraj uses “Be y’s first!” Be wise first!  meaning always start with the y vales on top (in the numerator)

TWO WAYS OF CALCULATING on a graph:

       1) Pick 2 points and use the following formula
Difference of the 2 y –values/ Difference of the 2 x-values
The formal is restated with SUBSCRIPTS on the x’s and y’s below: (memorize this) y₂ – y₁/ x₂-x₁  The subscripts just differentiate between point one and point two. You get to decide which point is point one or two. I usually try to keep the difference positive, if I can—but often, one of them will be negative and the other will be positive.

EXAMPLE:   ( 3, 6)  and (2, 4)    y₂ – y₁/ x₂-x₁       6-4/3-2 = 2/1 = 2

    2)Count the slope on the GRAPH using rise over run.
From the point (2,4) count the steps UP ( vertically) to (3,6): I get 2 steps
Now count how many steps over to the right (horizontally): 1 step
Rise = 2 and Run = 1 or 2/1 = 2

HORIZONTAL LINES  have only a y intercept (unless it’s the line y = 0 and then that is the x-axis) The equation of a horizontal line is y = b where b is a constant. Notice that there is NO X in the equation. For example y = 4 is a horizontal line parallel to the x-axis where the y value is always 4 What is the x value? All real numbers! Your points could be ( (3, 4) or ( 0, 4) or ( -10, 4)
Notice y is always 4! The constant rate of change  or slope is 0

If you take any 2 points on a horizontal line the y values will always be the same so the change ( or difference) in the numerator = 0.

EXAMPLE  y = 4

Pick any two points Let’s us ( 3,4) and (-10, 4)

(4 - 4)/ (3 - ^-10) becomes ( 4-4)/ 3 + 10 = 0/13 = 0

VERTICAL LINES ( which are NOT functions)  have only an x intercept ( unless it is the line x = 0 and then it is the y-axis) The equation of a vertical line is x = a, where a is a constant. Notice that there is NO Y in this equation.

EXAMPLE: x = 4

This is a vertical line parallel to the y axis 4 steps to the right of it. Pick any two points on this line Let’s use ( 4, -1) and (4, 7)

This time the change in y is -1  - 7 = -8

and the change in x is 4 -4 = 0

BUT -8/0 is UNDEFINED

Make sure you write undefined for the slope!

Finding a Missing Coordinate if you know 3 out of 4 values and the Slope

Say you know the following:
(1,4) and (-5, y) and the slope is given as 1/3

Find the missing y value

Use the slope formula

Change in y/ change in x

(y – 4)/- 5 – 1  and you know that the slope is 1/3

That means

(y – 4)/- 5 – 1   = 1/3

(y – 4)/-6 = 1/3

Solve

3(y -4)= -6

3y – 12 = -6

 y = 2

Or you could have divide both sides by 3 FIRST

y - 4 = -2

y = 2