Algebra Honors ( Periods 6 & 7)

Direct & Inverse Variations  8-9 and 8-10

Direct Variations  8-9

 f(x) = mx + b

It is a linear function. the f(x) is dependent on the x value.

A direct variation is a function defined by an equation in the form

y = kx, where k is a non zero constant. When graphing, k is  the slope

You can say that y varies directly as x

that is, as x goes up in value so does y or

as x decreases à so does y

When the domain is the set of all real numbers, the graph of a direct variation is a straight line with slope k that passes through the origin.

Given that m varies directly as n, and that m= 42 when n = 2, the constant of variation can be found by writing m = kn and substituting in those values. You find that

42 = 2k , so k = 21.
You can write the function as m = 21n.
Now when asked to find the value of m when n = 3 you just plug and chug…
m = 21(3)  so m = 63

Could we have found this a different way? YES..

Suppose ( x₁, y₁) and  ( x₂, y₂) are two ordered pairs of a direct variation defined by

y = kx and you know that neither  x₁ nor  x₂  are zero.

You know that

y₁ = kx₁    and that y₂= kx₂ 

Solving both for k you discover that

y₁/x₁ = k and so does y₂/x₂ = k

Since each ratio equals k, the ratios are equal and you can set them in an equation

y₁/x₁ = y₂/x₂  and you read this… “y₁ is to x₁ as y₂ is to x₂.”

This is a proportion!!

For this reason k is sometimes called the constant of proportionality  and
y is said to be directly proportional to x

When you use a proportion to solve a problem, you will want to recall that the product of the extremes equals the product of the means.

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 variations:

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

Inverse Variations 8-10

An inverse variation is a function defined by an equation in the form:

xy = k where k is a NON ZERO constant

or  y = k/x  where x ≠ 0

You say that y varies inversely as x or that y is inversely proportional to x.

The constant k is the constant of variation.

The graph of an inverse function is NOT a straight line!

xy = k is NOT linear!

Your graph can be a hyperbola

When k is positive the branches of the graph are in Quadrants I and III

When k is negative the branches of the graph are in Quadrants II and IV

Similarly to direct variation, you can compare two ordered pairs of the same inverse variation. Since the coordinates must satisfy the equation xy = k you know that

x₁y₁=k and x₂y₂= k

or

x₁y₁= x₂y₂

Reviewing :

Direct Variation is y = kx    or y₁/x₁ = y₂/x₂

Inverse Variation  is xy = k or x₁y₁= x₂y₂

These equations show that for direct variation the quotients of the coordinates are constant and for inverse variation the products of the coordinates are constant.

Is it Direct or Inverse Variation?

y/x = k  (careful this becomes   y = kx    so its direct)

y= k/x           inverse

p = k/z          inverse

xy = 25         inverse

d= 40t          direct

m/n= 5/8      direct

x/y = 1/k      direct

kxy = 5        inverse

WORD PROBLEMS FOR DIRECT & INVERSE VARIATIONS ( 2^nd day of lesson)

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 last year… remember those winds…

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

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

Math 6A ( Periods 1 & 2)

Scale Drawing 7-9

Using the book’s drawing on page 237, find the length and width of the room shown, if the scale of the drawing is 1cm: 1.5 m

Measuring the drawing, we find that it has a length of 4 cm and a width of 3 cm
Method 1: write a proportion for the length
Let l = the actual length
1/1.5 = 4/ l
l= 4 (1.5)
l = 6
The room is 6 m long
Write a proportion for the width
Let w = the actual width
1/1.5 = 3/w
w = 3 (1.5)
w = 4.5 m
The room is 4.5 m wide

A scale drawing is a diagram of an object where its length and width aare proportional to the actual length and width.

The scale for the drawing gives the relationship between the drawing's measurements and the actual measurements.
For example 1 inch represents 2 feet

The Scale Factor is the ratio of the length in the drawing to the corresponding actual lenth.

A Scale Factor on a matchbox car of 1/64 would mean that every part of the matchbox car was 1/64 of the actual car-- the windshield, the door, the wheels, the roof... would all have the same relationship.

Scale Factor MUST HAVE the same units of measurements.



Method 2 : Use the scale ratio

1 cm/ 1.5 m = 1/cm/150 cm = 1/150
You need to change the units to the same and then set up a ratio

The actual length is 150 times the length in the drawing so
l =150 (4) = 600 cm = 6 m
w =150 (3) = 450 cm = 4.5 m

The scale on a map is 1 cm to 240 m


Open your books to Page 237, you will notice a drawing of a house. In this drawing of the house, the actual height of 9 meters is represented by a length of 3 centimeters, and the actual length of 21 meters is represented by a length of 7 centimeters.
This means that 1 cm in the drawing represents 3 m in the actual building. Such a drawing in which all lengths are in the same ratio to actual lengths is called a scale drawing.
The relationship of length in the drawing to actual length is called the scale. In the drawing of the house the scale is 1cm: 3m

We can express the scale as a ratio, called the scale ratio, if a common unit of measure is used. Since 3 m = 300 cm, the scale ratio is 1/300

The distance from Ryan’s house to his school is 10 cm on the map. What is the actual distance?
let d = the distance from Ryan’s house to school
1/240 = 10/d so d = 240(10)
d = 2400m or
The distance from Ryan's house to school is 2.4 km


A picture of an insect has a scale 7 to 1. The length of the insect in the picture is 5.6 cm. What is the actual length of the insect in millimeters ?
Let a= the actual length of the insect in cm

7/1 = 5.6/ a
7a = 5.6
a = .8 cm

or 7/a = 56/
The actual length of the insect is 0.8 cm which is 8 mm

OR change the given 5.6 cm into 56 mm to begin with!
this time
let a = the actual length of the insect in mm
7/1 = 56/a
Now we can simplify easily
a = 8
so
the actual length of the insect is 8 mm

Algebra Honors ( Periods 6 & 7)

Linear and Quadratic Functions 8-8

This lesson will be completed in two parts:

Linear Functions:

The function g defined by g(x) = 2x - 3  is called a linear function

Notice the word line in linear

If its domain is the set of all real numbers  then the straight line that is the graph of

y = g(x) = 2x - 3  is the graph of g . 

The slope of the graph is 2 and the y-intercept is -3.

A function f defined by f(x) = mx + b is a linear function

If the domain of f is the set of all real numbers, then its graph is the straight line with slope m and y-intercept b.

Relations:

A relation is any set of ordered pairs.

The set of the first coordinates of the ordered pairs is the domain of the relation

The set of the second coordinates of the ordered pair is the range.

A function is a relation in which different ordered pairs have different FIRST coordinates.

Math 7 (Period 4)

Multiplying and Dividing Powers 6.7

 We are discovering the basic multiplication rules for powers having the same bases.

Multiplying Powers

a⁵ ∙a³= a∙a∙a∙a∙a∙a∙a∙a = a⁸

 RULE….to multiply two powers with the SAME BASES just add their exponents

a^m ∙ aⁿ = a^m+n

3²∙3³ = 3⁵

( -½)² (-½ )  = (-½ )³

Multiplygin Variable Expressions:

2x²∙3x⁴ = (2∙3)(x²∙x⁴) = 6x²⁺⁴ = 6x⁶

Dividing Powers

c⁵/c³

c∙c∙c∙c∙c
c∙c∙c

= c∙c

= c²

c⁵/c³ = c^5-3 = c²

To divide two powers with the SAME BASES just subtract the exponents

a^m/aⁿ = a^m-n

2⁷/2⁴= 2^7-4 = 2³

(-5)³/-5

Careful with this one… it is really

(-5)³/(-5)¹

Now they are the same base so it is just a subtraction problem with the exponents

(-5)^3-1 =(-5)² = (-5)(-5) = 25

So let’s look at two examples

(-2)⁵ means (-2)(-2)(-2)(-2)(-2) = -32

But

(-2)⁴ means  (-2)(-2)(-2)(-2) = + 16

(-2)¹¹ à negative

(-2)¹⁰ à positive

***Odd/Even Rule with powers***

If there is a negative inside the parentheses

odd # of negative signs or odd  power à it will be negative

even # of negative signs or even power à it will be positive

(-4) ⁵⁵ à negative

(-4)⁵⁶ à positive

(-1)¹⁵ = -1

(-1)¹⁰⁰ = +1

If there is a negative but NO parentheses (NO HUGS!!)

such as -2⁵ or -2⁴

….IT will ALWAYS be negative

-2⁵ means take the opposite of 2⁵

and -2⁴ means take the opposite of 2⁴

-2⁵ = -2∙2∙2∙2∙2 = -32

-2⁴ =   -2∙2∙2∙2 = -16

(-10)⁴/(-10)² = (-10)^4-2 = (-10)²

Be careful

-10² ≠ (-10)²

-100 ≠ 100

(2/5)⁶
(2/5)³

= (2/5)^6-3

=8/125

Math 6A ( Periods 1 & 2)

Problem Solving: Using Proportion 7-8

Proportions can be used to solve word problems. Use the following steps to help you in solving problems using proportions
~ Decide which quantity is to be found and represent it by a variable
~ Determine whether the quantities involved can be compared using ratios (rates)
~ Equate the ratios in a proportion
~ Solve the proportion

Mrs. Nelson bought 4 tires for her car ( an Infiniti- what else would a math teacher own!) at a cost of $264. How much would 5 tires cost at the same rate?
Let c = the cost of 5 tires. Set up a proportion

4/264 = 5/c
Solve the proportion
4c = 5(264)
Now divide both sides by 4
4c/4 = 5(264)/4
c = 330
Therefore, 5 tires would cost $330

Notice, the proportion in this example could also have been written as

2/264 = c/5
In fact--Any of the following proportions can be used to solve the problem

4/5 = 264/c 5/4 = c/264 4/264 = 5/c 264/4 = c/ 5
All of the above proportions result in the same equation

4c = 5(264)
But be careful you need to use a proportion that does relate.
4/5 DOES NOT EQUAL c/264. That is not an accurate proportion and would result in an inaccurate solution.

tires ___________ = _____________tires
cost _____________=_____________cost


If 4 bars of soap cost $1.50, how much would 6 bars of soap cost?

let x = the cost of 6 bars of soap

bars ___________ = _____________bars
cost _____________=_____________cost
4/1.50 = 6/x
simplify first up and down ( BUT NEVER DIAGONALLY!!)
x = 3 (.75)
x = 2.25
$2.25
6 bars would cost $2.25


If 9 kg of fertilizer will feed 300 m² of grass, how much fertilizer would be required to feed 500 m² of grass?

kg ___________ = _____________kg
m² _____________=_____________m²

9/300 = n/500
SIMPLIFY FIRST
9/3 = n/5
3/1 = n/5
n = 15
15kg of fertilzer


For every 5 sailboats in a harbor, there are 3 motorboats. If there are 30 sailboats in the harbor, how many motorboats are there?

Let m = the number of motorboats
5/3 = 30/m

5m = 3(30) divide both sides by 5 5m/5 = 3(30)/5


m=18
There are 18 motorboats in the harbor.

Some guidelines you can use to determine when it is appropriate to use a proportion to solve a word problem.

Ask the following questions

If one quantity increase does the other quantity also increase? (If one quantity decreases, does the other quantity decrease?) When the number of tires is increase, the cost is also increased.

Does the amount of change (increase or decrease) o one quantity depend upon the amount of change (increase or decrease) of the other quantity? The amount of increase in the cost depends upon the number of additional tires bought.

Does one quantity equal some constant times the other quantity? The total costs equals the cost of one tire times the number of tires. The cost of one tire is constant.

If the answers to all the questions above is YES, then it is appropriate to use a proportion.

Sometimes setting up a table can be useful


Although the problems in this lesson may be solved without using proportions, I must insist that you write a proportion for each problem and solve using this method. You may check your work using another other method you know.