chap1e.html

Chapter 1: Lines, Slopes,Circles

Straight Lines

Much of the mathematics in this chapter will be a review for you.

In the ( ) coordinate system we normally write the -axis horizontally, with positive numbers to the right of the origin, and the -axis vertically, with positive numbers above the origin. That is, unless stated otherwise, we take ``rightward'' to be the positive x-direction and ``upward'' to be the positive -direction. We normally choose the same scale for the and -axes. For example, the line joining the origin to the point ( ) makes an angle of with the -axis (and also with the -axis).

But in applications, more descriptive variable names are used, and often different scales are chosen along the two axes. In fact, the quantities represented by the two variables can be so different that the same scale has no meaning! For example, suppose you drop something from a window, and you want to study how its height above the ground changes from second to second. It is natural to let the letter t denote the time (the number of seconds since the object was released) and to let the letter s denote the height in meters. For each t (say, at one-second intervals) you have a corresponding height s. This information can be tabulated, and then plotted on the (t,s) coordinate plane. Here is an example:

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We use the word `` quadrant '' for each of the four regions the plane is divided into: the 1st quadrant is where points have both coordinates positive, and the 2nd, 3rd, and 4th quadrants are counted off counterclockwise as follows:

Suppose we have two points and in the ( )-plane. We often want to know the change in x-coordinate (also called the ``horizontal distance'') in going from A to B. This is often written x, where the meaning of is ``change in'' (thus, x can be read as ``change in '' --- it denotes a single number, and should not be read as ``delta times x''). In our example, = Similarly, the ``change in y'' is written y. In our example, y = 3-1=2, the difference between the y-coordinates of the two points. It is the vertical distance you have to move in going from A to B. The general formulas for the change in x and the change in y between a point ( ) and a point ( ) is:

x = , y = .

Note that ``the change'' is algebraic. For example the values of x and y in going from B to A will be respectively -1 and -2. You should follow the above formula all the time.

If we have two points and , then we can draw one and only one line through both points. By the slope of this line we mean the ratio of to x. The slope is often denoted m: m= = . For example, the line joining the points A and B in the last paragraph has slope 2.
Note that this slope does not depend on which two points on the line are taken or on the order in which they are taken . For instance, if we take the change from B to A, then we will get the ratio of which still evaluates to 2.

There is a very simple procedure to find all points on the line joining and . Any point P(x,y) on the line has coordinates { , where t is any real number}.

Note : This way of describing the points on the line between A and B can be represented symbolically as . Here and we add points by adding their coordinates.

Convince yourself (with pencil and paper !) that t=0 gives the point A, t=1 gives the point B and values of between give the line segment from to B. You should investigate how the different values of t correspond to various subdivisions of the segment . Also verify once and for all that the slope is the same if we choose the points A,B or A,P, regardless of the value of t.

The importance of thinking of lines this way cannot be overemphasized since when we want to represent lines in dimensions other than two (e.g. in three dimensions) then this works while there is no longer a simple "point-slope" form. For instance if B(3,-2,4) and A(-1,5,7) are points in space then the points on the line connecting them are exactly the points

where ranges over all real numbers. As before gives , gives , gives the point midway between the points and and gives the point of the way from to . You can easily prove this by calculating the distance from the point corresponding to from and . If the assertion is correct then one of these should be twice the other and their total should be the distance from to .

Example 1.1: Polygonal Lines

According to the 1990 U.S. federal income tax schedules, a head of household pays 15% on income up to $26,050. If the taxable income is between $26,050 and $134, 930, then, in addition, 28% must be paid on the amount between $26,050 and $67,200, and 33% paid on the amount over $67,200 (if any). Interpret the tax bracket information (15 %, 28 %, or 33 %) using mathematical terminology, and graph the tax on the y-axis against the taxable income on the x-axis.

Solution

The percentage, when converted to a decimal (0.15, 0.28, and 0.33), is the slope of the straight line which is the graph of tax for the corresponding tax bracket. The tax graph is what's called a polygonal line , i.e., it's made up of several straight line segments of different slopes.

Functions whose graphs are polygonal lines are called piecewise linear functions. In the tax graph the first line starts at the point (0,0) and heads upward with slope 0.15 (i.e., it goes upward 15 for every increase of 100 in the x-direction), until it reaches the point above x=26050. Then the graph ``bends upward,'' i.e., the slope changes to 0.28. As the horizontal coordinate goes from x=26050 to x=67200, the line goes upward 28 for each 100 in the x-direction. At x=67200 the line turns upward again, and continues with slope 0.33. See the following graph.

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Slopes

The most familiar form of the equation of a straight line is: . Here m is the slope of the line: if you increase by , the equation tells you that you have to increase by . If you increase x by , then y increases by . The number is called the y-intercept , because it is y-coordinate where the line crosses the y-axis.

If you know two points on a line, the formula gives you the slope. Once you know a point and the slope, then the y-intercept can be found by substituting the coordinates of the point in the equation: , i.e., . Alternatively, one can use the `` point-slope '' form of the equation of a straight line, which is: . This relation says that ``between the point ( ) and any other point ( ) on the line, the change in y divided by the change in x is the slope m of the line.''

In general the equation of a line can be expressed as a linear relation of the form where at least one of the coefficients are nonzero. You should convince yourself that every possible line has such an equation and conversely such an equation always gives a line for its graph.

A convenient way of writing the final equation of a line thru two points ( ) and ( ) is:

( )( )=( )( )

You should verify that this is correct by checking that the equation is satisfied by both the points and that it has the correct form for the equation of a line. This is enough justification! For example, if we want to find the equation of the line joining our earlier points and , we can use this formula:

so that , i.e. .

The slope m of a line in the form tells us the direction in which the line is pointing. If is positive, the line goes into the quadrant as you go from left to right. If m is large and positive, it has a steep incline; while if is small and positive, then the line has a small angle of inclination. If is negative, the line goes into the quadrant as you go from left to right. If is a large negative number (large in absolute value), then the line points steeply downward; while if is negative but near zero, then it points only a little downward. These four possibilities are illustrated in the following graphs.

code for graphs above

If , then the line is horizontal: its equation is simply .

There is one type of line that cannot be written in the form , namely, a vertical line . A vertical line has an equation of the form . Sometimes one says that a vertical line has an ``infinite'' slope.

Sometimes it is useful to find the x-intercept of a line . This is the x-value when . Setting equal to and solving for
gives: . For example, the line through the points and has x-intercept Convince yourself that a vertical line like has x-intercept and has no y-intercept. The special vertical line coincides with the y-axis and may be said to have infinitely many y-intercepts.

Example 1.2: Slopes and Intercepts

Suppose that you are driving to Seattle at a constant speed, and notice that after you have been traveling for 1 hour (i.e., ), you pass a sign saying miles to Seattle; and after driving another half-hour you pass a sign saying miles to Seattle. Using the horizontal axis for the time t and the vertical axis for the distance y from Seattle, graph and find the equation for your distance from Seattle. Find the slope, y-intercept, and t-intercept, and describe the practical meaning of each.

Solution

The graph of y versus t is a straight line, because you are traveling at constant speed. The line passes through the two points ( ) and ( ). So its slope is = . The meaning of the slope is that you are traveling at a velocity of mph. The slope m is negative because you are traveling toward Seattle, i.e., your distance y is decreasing. The word ``velocity'' is often used for , when we want to indicate direction, while the word ``speed'' refers to the magnitude (absolute value) of velocity, which is mph. To find the equation of the line, we use the point-slope formula:

, so that so that .

The meaning of the y-intercept is that when t=0 (when you started the trip) you were miles from Seattle. To find the t-intercept, set , so that = . The meaning of the t-intercept is: the time when you'll be in Seattle. After traveling hrs min, your distance from Seattle will be .

Distance Between Two Points; Circles

Given two points ( ) and ( ), recall that their horizontal distance from one another is , and their vertical distance from one another is . The actual distance from one point to the other is the hypotenuse of a right triangle with legs and :

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The Pythagorean theorem then says that the distance between the two points is the square root of the sum of the squares of the horizontal and vertical sides:

distance = = .

For example, the distance between our two points and is equal to

= 2.236 ...

Remark ( general distance formula ) : This formula (properly interpreted) works in all dimensions. For instance if A=( ) and B=( ) are points in space then the distance from to is

distance(A,B) =

You should be able to write down (and calculate with) the formula for the distance between two points in four space, five-space, etc.

As a special case of the distance formula, suppose we want to know the distance of a point (x,y) to the origin. According to the distance formula, this is equal to .

A point ( ) is at a distance from the origin if and only if , or, if we square both sides: . This is the equation of the circle of radius r centered at the origin . The special case is called the unit circle its equation is . Similarly, if is any fixed point, then a point ( ) is at a distance r from the point C if and only if , i.e., if and only if