Direct & Inverse Variations 8-9 & 8-10
f(x) = mx + b
It is a linear function. the f(x) is dependent on the x value.
Several examples were given in class:
Truck rental Company charges $35 a day plus 21 cents per mile. Normally your questions in the past were "What is the cost for a rental of a truck for ...:
1 day and 340 miles?" or
"2 days and 450 miles?"
You would just plug in and figure out the exact cost...
Then we discussed the rental of a chain saw ( from the textbook)... remember it is for cutting down trees... like those which were knocked down by those tremendous winds we had a few months back...
We are given that the rental is $5.90 a hour and you must pay $6.50 for 1 can of gas. Again, in the past your questions would be something like...
"How much would it cost for 7.5 hours?"
But... you could write a linear function to represent the cost for all different rental hours.
Let h represent the hours
s(h) = 5.9h + 6.5
Now, no matter how many hours you rent the chain saw, you can figure out the cost.
Phone bills in the past charged 15 cents per message + a base charge. Let's say you were given the July bill of $18 which included 62 messages. What was the base charge?
First find out the cost of the messages (.15)(62) = $9.30 and subtract that from $18.
Or just realize it would be 18 - (.15)(62) = $ 8.70
Then you could write a linear function p(x) = .15x + 8.7
August had 76 messages... what was the bill becomes easy to solve-- just use the linear equation.
p(76) = .15(76) + 8.7 = 20.10. August bill was $20.10
Direct Variation
Direct variation is a function ( abbreviate fcn) defined by the following equation:
y = kx where k is a non-zero constant
y varies directly as x
k is called the constant of variation
( In graphing k represents the slope)
Example:
m varies directly as n
m = 42 and n= 2
Find:a) the constant of variation
b) value of m when n = 3
let m = kn
Finding: a) the constant of variation
42= k(2)
42 = 2k
solve this one step equation
k = 21
finding b) value of m when n = 3
just substitute in
m = (21)(3)= 63
Could we have found this a different way? YES..
notice we have an ordered pair (n,m)
Think about it first in terms of x and y
That is, each pair would be (x1, y1) or (x2, y2)
if y =kx we have y1= kx1 and y2= kx2 which means that
y1)/x1 = k
and
y2)/x2 = k
so we can set the two equal to each other
y1)/x1= y2)/x2
k is the constant of proportionality
y is directly proportional to x... and we can solve using proportions
So... back to the question
m varies directly as n
m = 42 and n= 2
Find:a) the constant of variation
b) value of m when n = 3
42/2 = x/3
solving using your knowledge of proportions... from 6th grade..
x = 6
We reviewed a few equations and found the following to be direct variations:
y = 3x
p = 9s
d = 3.3t
even y/x = -5
But the following were determined NOT to be direct variatins:
y = 3x2
xy = 4
Eample:
y varies directly as x
y = 6 and x = 72 Find the constant of variation
y = kx
6 = k(72)
6 = 72k
k = 1/12
Turn to Page 394
#20
distance on a map varies directly to actual distance
m= distance on the map
d= actual distance
m = kd
Given that 1 in on the map ---> 10 miles
1 = k(10)
1 = 10k
k = 0.1
so formula is m = 0.1d
writing as a proportion you would have
1/10 = m2 / d2
# 22 Volume directly proportional to temp T in Kelvin
5 Liters 300 degrees
V = kT
5= 300k
k = 1/60
so formula is V = (1/60)T
and as a formula before you simplify
5/300 = V2 /T2
or 1/60 = V2 /T2
Friday, February 10, 2012
Thursday, February 9, 2012
Tuesday, February 7, 2012
Algebra Honors (Period 6 & 7)
Functions Defined by Equations 8-7
A relation is a set of ordered pairs such as
{ (2,3), (3,5), (-4, 0), (5, 0)}
it is also a function because no repeating of the x value,
Domain of a relation is the set of 1st coordinates
the x values
Range is the 2nd coordinates
the y values
So in the example above
Domain is {2, 3, -4}
Range {3, 5, 0}
A relation that assigns to each value in the domain exactly one value in the range is called a FUNCTION
{ (2,3), (3,5), (-4, 0), (5, 0)} is a FUNCTION, whereas,
{ (2,3), (2,5), (-4, 0), (5, 0)} is NOT a function
values of domains ( x's) each are paired with only one element in the range.
Several ways to check... we looked at T- tables to compare the x values, we looked at mapping and we looked at graphs. Notice the vertical line test.
WE then compared the notation for functions
first we looked at y = 3x + 4 vs f(x) = 3x + 4
solve for x = 5
For y = 3x + 5
y = 3(5) + 4
y= 15 + 4
y = 19
Now, what was x again
Oh yeah... x = 5
so the ordered pair is (5, 19)
With f(x) = 3x + 4 however we have
f(5) = 3(5) + 4
f(5) - 15 + 4
f(5) = 19
and you can see what x was originally
(5, 19) is the ordered pair
a
Two ways to show functions
f(x) ... and we noted tht it could be g(x) or h(t) etc
or
f:x-->
g:x-->4 + 3x - x2
if the domain D = { -1, 0, 1, 2}
g:-1--> g(-1) = 4 + 3(-1) - (-1)2= 0
g:0-->g(0) = 4 +3(0) -0 = 4
g:1--> g(1) 4 + 3(1) - (1)2 = 6
g:2-->4 + 3(2) - (2)2
Range = {0. 4. 6} Notice that we list each number once ( even if there is a repeated number)
f:x --> x2 - 2x for the set of all REAL numbers
find
f(4) = (4)2 -2(4) = 8
f(-3) = (-3) 2 -2(-3) = 15
f(2) = (2)2 -2(2) = 0
so the Range is {8, 15, 0}
Relations & Functions
RELATIONS: Set of ordered pairs where the x values are the DOMAIN and the y values are the RANGE.
FUNCTIONS: Relations where there is just one y value for each x value IN OTHER WORDS----YOU CAN'T HAVE TWO y VALUES for the SAME x value!!!
If you see x repeated twice, it's still a relation, but it's not a function.
In the real world, there are excellent examples....pizza prices.
A restaurant can't have two different prices for the same size cheese pizza.
If you charge $10 and $12 on the same day for the same pizza, you don't have a function.
But, you certainly can charge $10 for a cheese pizza and $12 for a pepperoni pizza.
VERTICAL LINE TEST: When you graph a function, if you draw a vertical line anywhere on the graph, that line will only intersect the function at one point!!!!
If it intersects at 2 or more, it's a relation, but not a function.
So a horizontal line function, y = 4, is a function, but a vertical line function, x = 4 is not.
Any line, y = mx + b, is a function.
INPUTS: x values
OUTPUTS: y values
f(x) means the value of the function at the given x value
You can think of f(x) as the y value
Finding the value of a function: Plug it in, plug it in!
f(x) = 2x + 7
Find f(3)
f(3) = 2(3) + 7 = 13
The function notation gives you more information than using y
If I tell you y = 13 you have no idea what the x value was at that point
But if I tell you f(3) = 13, you know the entire coordinate (3, 13)
Domain of a function = all possible x values (inputs) that keep the solution real
Range of a function = all possible y values (outputs) that result from the domain
EXAMPLE:
f(x) = x + 10 has the domain of all real numbers and the same range because every value will keep the answer f(x) a real number
EXAMPLE:
f(x) = x2 has the domain again of all real numbers, BUT the range is greater than or = to zero
because when a number is squared it will never be negative! So f(x) will always be 0 or positive
EXAMPLE:
f(x) = absolute value of x has the domain of all real numbers, but again the range will be greater than or equal to zero because absolute value will never be negative
EXAMPLE:
f(x) = 1/x has a domain of all real numbers EXCEPT FOR ZERO because it would be undefined if zero was in the denominator. The range is all real numbers except zero as well.
This function will approach both axes but never intersect with them.
The axes are called asymptotes which means that they will get very close but never reach them
EXAMPLE:
f(x) = (x - 10)/x + 3
Domain is all real numbers EXCEPT -3 because -3 will turn the denominator into zero (undefined)
What is the range?
A relation is a set of ordered pairs such as
{ (2,3), (3,5), (-4, 0), (5, 0)}
it is also a function because no repeating of the x value,
Domain of a relation is the set of 1st coordinates
the x values
Range is the 2nd coordinates
the y values
So in the example above
Domain is {2, 3, -4}
Range {3, 5, 0}
A relation that assigns to each value in the domain exactly one value in the range is called a FUNCTION
{ (2,3), (3,5), (-4, 0), (5, 0)} is a FUNCTION, whereas,
{ (2,3), (2,5), (-4, 0), (5, 0)} is NOT a function
values of domains ( x's) each are paired with only one element in the range.
Several ways to check... we looked at T- tables to compare the x values, we looked at mapping and we looked at graphs. Notice the vertical line test.
WE then compared the notation for functions
first we looked at y = 3x + 4 vs f(x) = 3x + 4
solve for x = 5
For y = 3x + 5
y = 3(5) + 4
y= 15 + 4
y = 19
Now, what was x again
Oh yeah... x = 5
so the ordered pair is (5, 19)
With f(x) = 3x + 4 however we have
f(5) = 3(5) + 4
f(5) - 15 + 4
f(5) = 19
and you can see what x was originally
(5, 19) is the ordered pair
a
Two ways to show functions
f(x) ... and we noted tht it could be g(x) or h(t) etc
or
f:x-->
g:x-->4 + 3x - x2
if the domain D = { -1, 0, 1, 2}
g:-1--> g(-1) = 4 + 3(-1) - (-1)2= 0
g:0-->g(0) = 4 +3(0) -0 = 4
g:1--> g(1) 4 + 3(1) - (1)2 = 6
g:2-->4 + 3(2) - (2)2
Range = {0. 4. 6} Notice that we list each number once ( even if there is a repeated number)
f:x --> x2 - 2x for the set of all REAL numbers
find
f(4) = (4)2 -2(4) = 8
f(-3) = (-3) 2 -2(-3) = 15
f(2) = (2)2 -2(2) = 0
so the Range is {8, 15, 0}
Relations & Functions
RELATIONS: Set of ordered pairs where the x values are the DOMAIN and the y values are the RANGE.
FUNCTIONS: Relations where there is just one y value for each x value IN OTHER WORDS----YOU CAN'T HAVE TWO y VALUES for the SAME x value!!!
If you see x repeated twice, it's still a relation, but it's not a function.
In the real world, there are excellent examples....pizza prices.
A restaurant can't have two different prices for the same size cheese pizza.
If you charge $10 and $12 on the same day for the same pizza, you don't have a function.
But, you certainly can charge $10 for a cheese pizza and $12 for a pepperoni pizza.
VERTICAL LINE TEST: When you graph a function, if you draw a vertical line anywhere on the graph, that line will only intersect the function at one point!!!!
If it intersects at 2 or more, it's a relation, but not a function.
So a horizontal line function, y = 4, is a function, but a vertical line function, x = 4 is not.
Any line, y = mx + b, is a function.
INPUTS: x values
OUTPUTS: y values
f(x) means the value of the function at the given x value
You can think of f(x) as the y value
Finding the value of a function: Plug it in, plug it in!
f(x) = 2x + 7
Find f(3)
f(3) = 2(3) + 7 = 13
The function notation gives you more information than using y
If I tell you y = 13 you have no idea what the x value was at that point
But if I tell you f(3) = 13, you know the entire coordinate (3, 13)
Domain of a function = all possible x values (inputs) that keep the solution real
Range of a function = all possible y values (outputs) that result from the domain
EXAMPLE:
f(x) = x + 10 has the domain of all real numbers and the same range because every value will keep the answer f(x) a real number
EXAMPLE:
f(x) = x2 has the domain again of all real numbers, BUT the range is greater than or = to zero
because when a number is squared it will never be negative! So f(x) will always be 0 or positive
EXAMPLE:
f(x) = absolute value of x has the domain of all real numbers, but again the range will be greater than or equal to zero because absolute value will never be negative
EXAMPLE:
f(x) = 1/x has a domain of all real numbers EXCEPT FOR ZERO because it would be undefined if zero was in the denominator. The range is all real numbers except zero as well.
This function will approach both axes but never intersect with them.
The axes are called asymptotes which means that they will get very close but never reach them
EXAMPLE:
f(x) = (x - 10)/x + 3
Domain is all real numbers EXCEPT -3 because -3 will turn the denominator into zero (undefined)
What is the range?
Math 6 Honors ( Periods 1, 2, & 3)
Ratios 7-6
In our textbook, the example given involves the number of students --at what I called a mythical middle school --as well as the number of teachers. There are 35 teachers and 525 students. We can compare the number of teachers to the number of students by writing a quotient
number of teachers
number of students
35
525
1/15
The quotient of one number divided by a second number is called the ratio of the first number to the second number.
We can write a ratio in the following ways:
1/15 OR 1:15 OR 1 to 15
All of these expressions are read one to fifteen.
If the colon notation is used the first number is divided by the second. A ratio is said to be lowest terms if the two numbers are “relatively prime.”
You do not change an improper fraction to a mixed number if the improper fraction represents a ratio
There are 9 players on a baseball team. Four of these are infielders and 3 are outfielders. Find each ratio in lowest terms.
a. infielders to outfielders
b. outfields to total players
# of infielders
# of outfielders
= 4/3 or 4:3 or 4 to 3
# of outfielders
# total of players
= 3/9 = 1/3 or 1:3 or 1 to 3
Some ratios compare measurements. In these cases we must be sure the measurements are expressed in the same units
It takes Kiana (or Helen or Emme) 4 minutes to mix some paint. It takes her 3 hours to complete painting her room. What is the ratio of the time it takes Kiana (or Helen or Emme) to mix the paint to the time it takes her to paint her room?
Use minutes as a common unit for measuring time. You must convert the hours to minutes first
3h = 3 • 60min = 180 min
The ratio is :
min. to mix
min. to paint
= 4/180 = 1/45 or 1:45
Some ratios are in the form
40 miles per hour or 5 pencils for a dollar
“ I want my… I want my…. I want my … MPG!!”
These ratios involve quantities of different kinds and are called rates. Rates may be expressed as decimals or mixed numbers. Rates should be simplified to a per unit form. When a rate is expressed in a per unit form, such a rate is often called a unit rate.
I know you will be driving in a blick of an eye... so Justin, in his Lamborghini(and Shane in his Corvette and Nick in his yellow Lamborghini) went 258 miles on 12 gallons of gas. Express the rate of fuel consumption in miles per gallon.
The rate of fuel consumption is
258 miles
12 gallons
= 21 1/2 miles per gallon
Some of the most common units in which rates are given are the following:
mi/gal or mpg miles per gallon
mi/h or mph miles per hour
km/L kilometers per liter
km/h kilometers per hour
Page 229
1 What is the cost of grapes in dollars per kilogram if 4.5 kg of grapes costs $7.56?
$7.56/4.5 kg divide carefully and you discover it is $1.68/kg
2. THe index of refraction of a transparent substance is the ratio of the speed of light in space to the speed of light in the substance.
Using the table from the textbook (look at page 229) Find the index of refraction of
a) glass
300,000/200,000 straight from the chart, which can simplify to 3/2
b) water
300,000/225,000 again from the chart, which can simplify to 4/3
3. The mechanical advantage of a simple machine is the ratio of the weight lifted by the machine to the forse necessary to lift it.
What is the mechanical advantage of a jack that lifts a 3200 pound car with a force of 120 pounds?
3200/120 = 80/3
4. The C string of a cello vibrates 654 times in 5 seconds. How many vibrations per second is this?
654 vibrations/5seconds... divide carefully and you find... 130 4/5 vibrations per second
5. A four-cubic-foot volume of water at sea level weights 250 pounds. What is the density of water in pound per cubic foot?
250 pounds/4 cubic ft ... divide carefully and you find 62 1/2 lb/ft3
6. A share of stock that costs $88 earned $16 last year. What was the price to earnings ratio?
88/16 = 11/2
7. we did in our spiral notebooks this year... please check
In our textbook, the example given involves the number of students --at what I called a mythical middle school --as well as the number of teachers. There are 35 teachers and 525 students. We can compare the number of teachers to the number of students by writing a quotient
number of teachers
number of students
35
525
1/15
The quotient of one number divided by a second number is called the ratio of the first number to the second number.
We can write a ratio in the following ways:
1/15 OR 1:15 OR 1 to 15
All of these expressions are read one to fifteen.
If the colon notation is used the first number is divided by the second. A ratio is said to be lowest terms if the two numbers are “relatively prime.”
You do not change an improper fraction to a mixed number if the improper fraction represents a ratio
There are 9 players on a baseball team. Four of these are infielders and 3 are outfielders. Find each ratio in lowest terms.
a. infielders to outfielders
b. outfields to total players
# of infielders
# of outfielders
= 4/3 or 4:3 or 4 to 3
# of outfielders
# total of players
= 3/9 = 1/3 or 1:3 or 1 to 3
Some ratios compare measurements. In these cases we must be sure the measurements are expressed in the same units
It takes Kiana (or Helen or Emme) 4 minutes to mix some paint. It takes her 3 hours to complete painting her room. What is the ratio of the time it takes Kiana (or Helen or Emme) to mix the paint to the time it takes her to paint her room?
Use minutes as a common unit for measuring time. You must convert the hours to minutes first
3h = 3 • 60min = 180 min
The ratio is :
min. to mix
min. to paint
= 4/180 = 1/45 or 1:45
Some ratios are in the form
40 miles per hour or 5 pencils for a dollar
“ I want my… I want my…. I want my … MPG!!”
These ratios involve quantities of different kinds and are called rates. Rates may be expressed as decimals or mixed numbers. Rates should be simplified to a per unit form. When a rate is expressed in a per unit form, such a rate is often called a unit rate.
I know you will be driving in a blick of an eye... so Justin, in his Lamborghini(and Shane in his Corvette and Nick in his yellow Lamborghini) went 258 miles on 12 gallons of gas. Express the rate of fuel consumption in miles per gallon.
The rate of fuel consumption is
258 miles
12 gallons
= 21 1/2 miles per gallon
Some of the most common units in which rates are given are the following:
mi/gal or mpg miles per gallon
mi/h or mph miles per hour
km/L kilometers per liter
km/h kilometers per hour
Page 229
1 What is the cost of grapes in dollars per kilogram if 4.5 kg of grapes costs $7.56?
$7.56/4.5 kg divide carefully and you discover it is $1.68/kg
2. THe index of refraction of a transparent substance is the ratio of the speed of light in space to the speed of light in the substance.
Using the table from the textbook (look at page 229) Find the index of refraction of
a) glass
300,000/200,000 straight from the chart, which can simplify to 3/2
b) water
300,000/225,000 again from the chart, which can simplify to 4/3
3. The mechanical advantage of a simple machine is the ratio of the weight lifted by the machine to the forse necessary to lift it.
What is the mechanical advantage of a jack that lifts a 3200 pound car with a force of 120 pounds?
3200/120 = 80/3
4. The C string of a cello vibrates 654 times in 5 seconds. How many vibrations per second is this?
654 vibrations/5seconds... divide carefully and you find... 130 4/5 vibrations per second
5. A four-cubic-foot volume of water at sea level weights 250 pounds. What is the density of water in pound per cubic foot?
250 pounds/4 cubic ft ... divide carefully and you find 62 1/2 lb/ft3
6. A share of stock that costs $88 earned $16 last year. What was the price to earnings ratio?
88/16 = 11/2
7. we did in our spiral notebooks this year... please check
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