Algebra (Period 4)

Fitting Equations to Data:7-7
(word problems)

Many real world relationships are LINEAR (or approximate a linear relationship), meaning they can be graphed with a LINE.
For example, if a candy bar costs $1.50, then 2 bars cost $3.00 etc


Think of the number of candy bars as the x value because that's the independent variable...
you decide how many candy bars you're going to buy.
The y value is the dependent variable.
How much money you owe at the register is the y value!
x y
1 $1.50
2 $3.00

3 $4.50

and so on....
If you graph this, you'll get a line because the price is constant.

That means that the slope is constant.


What is the slope????

The price!!! ($1.50)
Think of it as the change in y ($ you owe)
over the change in x (# of candy bars)


The money you owe goes up $1.50 every time you buy 1 more candy bar


This is a POSITIVE slope of 1.50!
So another meaning of a positive slope in the real world is that two types of data are changing at a constant rate IN THE SAME DIRECTION (called a POSITIVE CORRELATION in statistics)

You can reverse both directions and have the same relationship just as we did when we counted the slope going both up and down:

Every time you decrease your purchase by 1 bar, your purchase price also decreases by $1.50.


The data is still going in the SAME DIRECTION (now both going down!)


How do you find the equation of this linear relationship?
You use the Slope Intercept or Point Slope formulas of a line!

(It's just a real life mystery line!!! :)



SLOPE INTERCEPT:

y = 1.5x

(the y intercept is 0 because at 0 candy bars, you owe 0)

POINT SLOPE:

You can pick any point, which means any number of candy bars with its associated cost.
I used 1 candy bar (the x value) with its cost of $1.50:
y - 1.50 = 1.50(x - 1)


Y INTERCEPTS IN THE REAL WORLD THAT ARE NOT ZERO:

The scenario above has a y intercept value of 0 because you don't owe anything if you don't buy anything,
But often, the y intercept value will be a number.

For example, think of cell phone use.
Say you are charged $.10 per minute of use, but your monthly charge is $25.

Even if you don't use your cell phone, you still owe $25!

The linear equation would be:


y = .10x + 25 where y = what you owe
and x = number of minutes used
Plug in 0 for x, number of minutes, and you still owe $25 (y value)


NEGATIVE SLOPES IN THE REAL WORLD:
Let's think of a real life example that will give us a NEGATIVE SLOPE...

Often, the more you buy, the smaller the unit price per item.

This happens with copying or buying things like invitations.
Companies give you a "break" if you buy more.


SEE THE EXAMPLE ON PAGE 333 IN YOUR BOOK!



DATA THAT APPROXIMATES A LINEAR EQUATION:

Finally, sometimes real world data can be APPROXIMATED as a linear relationship.

In other words, it may not be exact, but a good way to understand the data is to look at it that way.


Think SCATTER PLOTS with POSITIVE or NEGATIVE CORRELATIONS.