MONT 106N – Identifying Patterns Group Discussion – Lines October 5, 2009

Background and Goals

After the Columbus Day break, we will be starting into a new section of the Freedman, Pisani, Purves text dealing with correlation and linear regression, two important statistical techniques aimed at identifying relationships between quantities. To deal with both of these ideas, we need to review a very basic topic from high school algebra and geometry: the various forms of equations of straight lines in the coordinate plane. This should be something you know well (or knew well at some point!). The purpose of today’s discussion is to “jog the memory.”

Discussion Questions

A. We all know that given two different points P = (x_{1}, y_{1}) and Q = (x_{2}, y_{2}) in the plane there is one and only one straight line containing both P and Q. What are two other geometric ways to specify a straight line uniquely? (One of these might be a special case of the other!)

B. Your answers to part A lead to two different ways to write down the xy-equation of the line. What are they, and what are the corresponding formulas? What constants are involved here, and what do they mean geometrically? (Notes: You can assume that the x-coordinates of P and Q are different here: x_{1} ≠ x_{2}. How does that make things simpler? What happens if x_{1} = x_{2}? How is that case different?)

C. Find an xy-equation for the line that is described in each part.

1. The line passing through the points P = (1, 2) and Q = (4, −1).

2. The line passing through the point P = (5, 3) and parallel to the line with equation y = 2x + 4.

3. The line passing through the point P = (5, 3) and perpendicular to the line from part 2.

4. The line passing through the y-axis at (0, 2) and crossing the x axis at an angle of 45^{∘} (measured from the x-axis to your line).

D. Suppose x and y are two quantities we can measure that are related in some way – changing the value of x changes the value of y. Think x = temperature measured in degrees Celsius and y = temperature measured in degrees Fahrenheit, for instance. We say that the two quantities x, y are (exactly) linearly related if the scatter plot of all measurements of x and corresponding measurements of y would show points along a single straight line.

1. Are F = temperature measured in degrees Celsius and C = temperature measured in degrees Fahrenheit linearly related? If so, what is the equation of the line that the (F, C) points lie along. (Recall that 32^{∘}F is the same as 0^{∘}C and 212^{∘}F is the same as 100^{∘}C.)

2. Today’s forecast high temperature in Worcester is 61^{∘}F. What is the equivalent temperature in degrees Celsius?

3. If you write your line from part 1 in the form C = mF + b, what do the constants m, b represent in “temperature terms” (what is their physical significance)?

4. Is there a value of F where C = F? If so, what is it?

Assignment

Group writeups due at the end of class.