Class Notes Precalculus I, 9/22/98

Symbolic Slopes; Slope = Slope Equation

We can find a straight line which we have fit to a set of data by choosing two points on the line and plugging their coordinates into the form y = mx + b of a linear function.

We wish to fit a straight line to the spring length vs. load data plotted on the graph below, and to determine the linear function for that line.

• Having sketched the line we select a pair of points, not too close together, from which to determine the equation of the line.
• Here we choose points (2,8) and (8,23).

We then substitute the coordinates of the points into the general form y = mx + b of a linear function.

• We obtain two equations in the unknowns m and b.
• Subtracting the two equations we eliminate b and obtain 15 = 6 m, which we solve for m.

• We obtain m = 2.5 (note the error in the figure: x = 2.5 should read m = 2.5).

• This value is substituted back into the second of the original equations to obtain an equation for b, which we solve.
• We substitute the values we have obtained for m and b into the y = mx + b form to obtain the equation y = 2.5 x + 3.

This form does not require us to remember what x and y stand for as we pose and answer questions regarding model.

• We begin by constructing the slope triangle and, to keep the symbolic quantities straight, labeling the positions a and b on the x axis and the corresponding positions f(a) and f(b) on the y axis.
• It is then clear that the run from the first point to the second is b - a, and that the rise is f(b) - f(a).
• We can thus write our desired result:

• This is in fact something that we would wish to do if it was possible.
• However the expression does not simplify, at least not without further information about the function f(x).
• Well-intentioned attempts to simplify revealed two bad ideas, discussed below.

• Looked at in this way it certainly looks like the distributive law applies so that f factors out and f(b) - f(a) = f(b-a).
• However, this is not correct.

• That this is not the case is easily illustrated by specifying almost any nonlinear function f(x) and choosing almost any two numbers for a and b.

• For example in the figure we choose f(x) = `sqrt(x), a = 25 and b = 16.
• Then if f(b) - f(a) = f(b-a), we would have f(25) - f(16) = f(25 - 16).
• For for this function we would thus have `sqrt(25) - `sqrt(16) = `sqrt(25-16).
• This simplifies to the statement 5 - 4 = 3.
• Since this is clearly false we see the error in assuming the distributive property for function notation.

Note that in the video clip it is stated that f(b) - f(a) = f(b-a) holds true for linear functions.

• This statement is false, and clearly illustrates that your instructor too can unthinkingly fall prey to bad ideas.
• You might pick a linear function whose y intercept is not 0 and test the instructor's statement to see why this is false.
• The statement what have been true if the instructor had restricted the statement to linear functions with y-intercept 0; you might check this out.

• This is the idea that (f(b) - f(a)) / (b-a) = f(b) / b - f(a) / a.
• We can substitute just about any numbers for a, b, f(b) and f(a) and see that this statement is false.
• There is in fact nothing at all in the properties of real numbers that justifies such a step.
• The first bad idea at least looked like the distributive law, this one looks like really bad order of operations, which is why this idea is worse than the former.

• In the figure below we begin with the points (1,4) and (8,9).
• We will make use of a simple but profound picture and the fact that the slope between any two points lying on a straight line is the same.

• If the point lies on the line then the equations satisfied.
• Otherwise the equation is not satisfied.

• We construct the slope triangle between (x, y) and one of the two given points.
• In the figure below we select the first point, and construct a slope triangle whose run is x - 1 and whose rise is y - 4.
• The slope between (1,4) and (x, y) is thus seen to be (y-4) / (x-1).

• To isolate y, we first multiply both sides by the denominator (x-1) of the left-hand side.
• We then note that (x-1) / (x-1) = 1, so that (y-4) / (x-1) * (x-1) = y-4, obtaining the third equation below.
• At this point we have a choice of adding 4 to both sides, which will accomplish our goal of isolating y, or first applying the distributive law to the right-hand side.
• Either choice is good. Someone suggested applying the distributive law first, so we follow this suggestion.

• It is true that in some senses decimals are easier to work with then fractions, and this is especially true for students who have grown up doing all their mathematics on a calculator and who have never really learned to work with fractions.
• However, in many applications small errors have a way of amplifying themselves into serious errors.
• The fractions involved in this problem, and most fractions that we encounter in applications, are expressed by calculators in decimal forms that have a slight round-off error.
• For this and other reasons it is important to be able to solve problems involving fractions.
• The process is somewhat tedious but not particularly difficult, and it also illustrates the use of the common-sense rules of algebra.

• Having arrived at the third step, where y - 4 = 5 / 7 * ( x - 1), this individual proceeded to multiply both sides of the equation by 7, correctly obtaining the indicated expression.
• However in the next step the right-hand side became 35 x - 35.
• When multiplying the expression in parentheses by 7, the individual forgot about the 7 in the denominator, which divides the 7 out front.
• So we see a very good idea and a bad idea expressed in one solution.

• Clearly the run is x2 - x1 and the rise is y2 - y1, so the slope is (y2 - y1) / (x2 - x1).
• If we locate an arbitrary point (x, y) on the line and construct its slope triangle with the point (x1, y1), we obtained a rise of y - y1 and a run of x - x1, and therefore a slope of (y - y1) / (x - x1).
• Setting the two slopes equal we have the slope = slope equation

• At some point in your mathematical career you have been asked to memorize this equation.
• The approach of this course is to begin with the simple picture we have used to develop the slope = slope equation and to proceed logically to obtain that equation.
• The ability to know where to start and to move logically from premises to conclusion is far more important then the ability to memorize and apply a meaningless equation.
• It is possible to associate a picture with the point-point form, but the picture is not as simple and elegant as the picture that goes with the slope = slope equation.
• You are therefore expected to be able to sketch the picture and proceed logically to the slope = slope equation.

• This is of course particularly valuable if we don't happen to know two points.
• If we know a single point (x1, y1) and the slope m then the slope = slope equation becomes (y - y1) / (x - x1) = m.

• We construct a single slope triangle between the known point (x1, y1) and the arbitrary point (x, y) on the line.
• We set the slope of this triangle equal to the known slope m, obtaining as before (y - y1) / (x - x1) = m.