Algebra Honors (Periods 5 & 6)

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

Algebra Honors ( periods 5 & 6)

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.