Algebra Honors (Period 6 & 7)

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