LINEAR EQUATIONS

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Instead of memorizing the formulas for Centigrade to Fehrenheit ----

why not use the fact that there exists
a linear relationship.
By forming the ordered pairs for

the Freezing and Boiling
Temperatures for WATER we get:

(0 , 32) and (100 , 212)

OR
(32 , 0 ) and (212 , 100)

Depending on which pair you use,

the slope will be either 9/5 or 5/9

CALCULUS - graphing f' and f"

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Comparing f to f' to f"

******************

Example:
If y = Parabola
then y ' = Straight Line

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f = position at time T

f ' = Velocity

f " = Acceleration

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If you are given the Velocity

and asked to find the Distance Traveled

from Time t1 to t2, then INTEGRATE

the Velocity from t1 to t2.

AREA UNDER the VELOCITY!

Sample Problems

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SLOPE (of a LINE)

FINDING the EQUATION of a LINE

What is this SLOPE THING?

Suppose we want to buy some T-shirts.

We are told that each T-shirt costs the

same amount. There is a charge

(an initial fee)

for setting up the logo for the order.

The price for 10 shirts would be $93
and 100 shirts would cost $633.

What is the COST for each shirt?

Dollars per Shirt!

(The SLOPE OF A LINE)

The difference in cost 633 minus 93 is $540.

$540 divided by 90 shirts gives

a RATE (Cost per Shirt) of $6.00 per shirt.

Why then do 10 shirts at $6 each cost $93?

Because there is a set-up fee of $33.

The Cost formula that Mr. Smith uses is:

Total Cost = ( $6 ) times # of shirts PLUS $33

We could represent this with the equation:

y = 6 x + 33

(Note: y depends on x for its value!)

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Take a look at the set of ordered pairs:

{ (3,11) (4,13) (5,15) (6,17)}

We notice that when X changes one unit

(as it does from 3 to 4), Y changes 2 units

(11 to 13). As X changes 2 units (3 to 5),

Y changes 4 units (11 to 15).

NOTE:

We are not saying that the values for y

are twice as big as the values for x.

We are talking about changes in y and x.

Y's change is 2 units for each 1 unit of change in x.
We call this the RATE OF CHANGE OF Y

WITH RESPECT TO X. Graphing these points

onto the xy-plane and connecting the points

with a line we have what is called a line

with SLOPE of 2. The EQUATION that these

points SATISFY is y = 2x + 5. Notice the

SLOPE of 2 is situated next to x.

Picking a point on the graph and moving

3 units to the RIGHT (RUN)

and then moving

6 units UP (RISE) we will land

us back onto the line.

The RISE is 6 and the RUN is 3.

the FRACTION (RATIO)

RISE/RUN EQUALS 2.

One definition for the slope of a LINE

is the RATIO

m = RISE/RUN.

How about the graph of the equation

y = (3/4)x - 1

Some of the ordered pairs that satisfy

this equation are {(4,2) (8,5) (12,8) ...}

When x changes 4 units, y changes 3 units.

The RATIO of 3 to 4 represents the

change in y with respect to the change in x.

THE CHANGE IN Y over the CHANGE in x is 3/4.

The slope of the graph of this equation is 3/4.

The SLOPE of the graph of the solutions to:

(i) a LINEAR FUNCTION f(x) = mx + b

(ii) an equation in the form Ax + By = C

(iii) or in the form y = mx + b

..........SLOPE is the rate that y changes

with respect to a change in x

..........In other words,

m = (y2-y1) / (x2-x1)

or m = change of y divided by the change in x

-----the values for x change by (x2-x1)

and the corresponding y's change by (y2-y1)

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Matrix Multiplication Example

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