College Algebra Notes, Section 1-5

Notes, Lesson 1.5
Models and Curve Fitting

Often when studying God's creation and data is collected, a linear relationship is approximated. An example is the graph below. This graph represents the magnitude (brightness) of heavenly bodies(X-Axis) plotted against their velocity (Y-Axis).

This plotting of data with two variables is called a scatter plot. It is called a scatter plot because in experiments, you have no idea how the data may be scattered. Above, even though the data has a definite pattern, there is some scattering.

For a reminder on how to use Excel to make a scatter plot, you could review this video lesson:

Scatter Plot with Excel Flash version

Anything that is calculated according to a linear formula will give an exact line. (With the exception of rounding off. Below you will find a set of ordered pairs which represent the grade chart for an assignment with 17 points. The ordered pairs are {(0,100),(1,94),(2,88), etc.}

# Wrong	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
Percent Correct	100	94	88	82	76	71	65	59	53	47	41	35	29	24	18	12	6	0

Here is the graph of the ordered pairs above. This too, is a scatter plot. Why is the graph not exactly straight?

Example Problem.
Using the data below, draw a scatter plot and find two prediction equations. Then predict the mileage for a $50,000 car.

Price in thousands	7.15	44.9	12.35	16.2	27.9	8.15	22	32.75	37	7.4	15.9	8.8
Miles/Gallon	41.8	13.3	23.8	20.3	25.0	35.4	24.4	21.8	12.3	31.0	28.0	25.9

Plot the data.

As is usually the case, this data does not form a perfect linear pattern. Where might we possibly draw the best-fit line?

To find prediciton equation #1, I chose the two points (7.4,31) and (22,24.4)

	Calculate the slope of the line containing the two points.
	Write the equation with the slope in place and the y-intercept missing.
	Pick one of the two points to substitute into the equation, and solve for b.
	Write the equation with the slope and y-intercept in place.
	Use the value of 50 ($50,000 car), to substitute into the equation and solve for y the mileage.

Pick two different points which also approximate the best-fit line. This time I picked (8.15,35.4) and (44.9,13.3).

	Calculate the slope of the line containing the two points.
	Write the equation with the slope in place and the y-intercept missing.
	Pick one of the two points to substitute into the equation, and solve for b.
	Write the equation with the slope and y-intercept in place.
	Use the value of 50 ($50,000 car), to substitute into the equation and solve for y the mileage.

Another approach to this problem is shown in this video lesson:

Linear Regression with Excel

Flash version

Some Notes from Class

Check Concepts

#1: A mathematical model is never a completely accurate reprentation of a physical situation - it is a(n) __________________.

#2: If there is no physical law or principle to help us formulate a model, we construct a(n) ____________ model, which is based entirely on collected data.

#3: True or False. A power function is another name for an exponential function.

#4: Plotting raw data points is called a _______________.

#5 In Microsoft's Excel, what command searches for a mathematical pattern in a series of plotted points?