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Analytic Geometry - Whitman College

1 Analytic GeometryMuch of the mathematics in this chapter will be review for you. However, the exampleswill be oriented toward applications and so will take some the (x, y) coordinate system we normally write thex-axis horizontally, with positivenumbers to the right of the origin, and they-axis vertically, with positive numbers abovethe origin. That is, unless stated otherwise, we take rightward to be the positivex-direction and upward to be the positivey-direction. In a purely mathematical situation,we normally choose the same scale for thex- andy-axes. For example, the line joining theorigin to the point (a, a) makes an angle of 45 with thex-axis (and also with they-axis).In applications, often letters other thanxandyare used, and often different scales arechosen in the horizontal and vertical directions. For example, suppose you drop somethingfrom a window, and you want to study how its height above the ground changes fromsecond to second.

160miles from Seattle. Tofind the t-intercept, set 0 = −50t+160, so thatt = 160/50 = 3.2. The meaning of the t-intercept is the duration of your trip, from the start until you arrive in Seattle. After traveling 3 hours and 12 minutes, your distance y from Seattle will be 0. Exercises 1.1. 1.

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Transcription of Analytic Geometry - Whitman College

1 1 Analytic GeometryMuch of the mathematics in this chapter will be review for you. However, the exampleswill be oriented toward applications and so will take some the (x, y) coordinate system we normally write thex-axis horizontally, with positivenumbers to the right of the origin, and they-axis vertically, with positive numbers abovethe origin. That is, unless stated otherwise, we take rightward to be the positivex-direction and upward to be the positivey-direction. In a purely mathematical situation,we normally choose the same scale for thex- andy-axes. For example, the line joining theorigin to the point (a, a) makes an angle of 45 with thex-axis (and also with they-axis).In applications, often letters other thanxandyare used, and often different scales arechosen in the horizontal and vertical directions. For example, suppose you drop somethingfrom a window, and you want to study how its height above the ground changes fromsecond to second.

2 It is natural to let the lettertdenote the time (the number of secondssince the object was released) and to let the letterhdenote the height. For eacht(say,at one-second intervals) you have a corresponding heighth. This information can betabulated, and then plotted on the (t, h) coordinate plane, as shown in figure use the word quadrant for each of the four regions into which the plane isdivided by the axes: the first quadrant is where points have both coordinates positive,or the northeast portion of the plot, and the second, third, and fourth quadrants arecounted off counterclockwise, so the second quadrant is the northwest, the third is thesouthwest, and the fourth is the we have two pointsAandBin the (x, y)-plane. We often want to know thechange inx-coordinate (also called the horizontal distance ) in going fromAtoB. This1314 Chapter 1 Analytic Figure data plot, height versus often written x, where the meaning of (a capital delta in the Greek alphabet) is change in.

3 (Thus, xcan be read as change inx although it usually is read as deltax . The point is that xdenotes a single number, and should not be interpreted as deltatimesx .) For example, ifA= (2,1) andB= (3,3), x= 3 2 = 1. Similarly, the change iny is written y. In our example, y= 3 1 = 2, the difference between they-coordinates of the two points. It is the vertical distance you have to move in going fromAtoB. The general formulas for the change inxand the change inybetween a point(x1, y1) and a point (x2, y2) are: x=x2 x1, y=y2 that either or both of these might be we have two pointsA(x1, y1) andB(x2, y2), then we can draw one and only one linethrough both points. By theslopeof this line we mean the ratio of yto x. The slopeis often denotedm:m= y/ x= (y2 y1)/(x2 x1). For example, the line joining thepoints (1, 2) and (3,5) has slope (5 + 2)/(3 1) = 7 to the 1990 federal income tax schedules, a headof household paid 15% on taxable income up to $26050.

4 If taxable income was between$26050 and $134930, then, in addition, 28% was to be paid on the amount between $26050and $67200, and 33% paid on the amount over $67200 (if any). Interpret the tax Lines15information (15%, 28%, or 33%) using mathematical terminology, and graph the tax onthey-axis against the taxable income on percentages, when converted to decimal values , , and , are theslopesof the straight lines which form the graph of the tax for the corresponding tax tax graph is what s called apolygonal line, , it s made up of several straight linesegments of different slopes. The first line starts at the point (0,0) and heads upwardwith slope ( , it goes upward 15 for every increase of100 in thex-direction), untilit reaches the point abovex= 26050. Then the graph bends upward, , the slopechanges to As the horizontal coordinate goes fromx= 26050 tox= 67200, the linegoes upward 28 for each 100 in thex-direction.

5 Atx= 67200 the line turns upward againand continues with slope See figure Figure vs. most familiar form of the equation of a straight line is:y=mx+b. Heremis theslope of the line: if you increasexby 1, the equation tells you that you have to increaseybym. If you increasexby x, thenyincreases by y=m x. The numberbis calledthey-intercept, because it is where the line crosses they-axis. If you know two pointson a line, the formulam= (y2 y1)/(x2 x1) gives you the slope. Once you know a pointand the slope, then they-intercept can be found by substituting the coordinates of eitherpoint in the equation:y1=mx1+b, ,b=y1 mx1. Alternatively, one can use the point-slope form of the equation of a straight line: startwith (y y1)/(x x1) =mandthen multiply to get (y y1) =m(x x1), the point-slope form. Of course, this may befurther manipulated to gety=mx mx1+y1, which is essentially the mx+b is possible to find the equation of a line between two pointsdirectly from the relation(y y1)/(x x1) = (y2 y1)/(x2 x1), which says the slope measured between the point(x1, y1) and the point (x2, y2) is the same as the slope measured between the point (x1, y1)16 Chapter 1 Analytic Geometryand any other point (x, y) on the line.

6 For example, if we want to find the equation ofthe line joining our earlier pointsA(2,1) andB(3,3), we can use this formula:y 1x 2=3 13 2= 2,so thaty 1 = 2(x 2), ,y= 2x course, this is really just the point-slope formula, except that we are not computingmin a separate slopemof a line in the formy=mx+btells us the direction in which the line ispointing. Ifmis positive, the line goes into the 1st quadrant as you go fromleft to large and positive, it has a steep incline, while ifmis small and positive, then theline has a small angle of inclination. Ifmis negative, the line goes into the 4th quadrantas you go from left to right. Ifmis a large negative number (large in absolute value), thenthe line points steeply downward; while ifmis negative but near zero, then it points onlya little downward. These four possibilities are illustrated in figure 4 2024 4 2 0 2 4 2024 4 2 0 2 4 2024 4 2 0 2 4 2024 4 2 0 2 4 Figure with slopes 3, , 4, and 0, then the line is horizontal: its equation is simplyy= is one type of line that cannot be written in the formy=mx+b, namely,vertical lines.

7 A vertical line has an equation of the formx=a. Sometimes one says thata vertical line has an infinite it is useful to find thex-intercept of a liney=mx+b. This is thex-valuewheny= 0. Settingmx+bequal to 0 and solving forxgives:x= b/m. For example,the liney= 2x 3 through the pointsA(2,1) andB(3,3) hasx-intercept 3 that you are driving to Seattle at constant speed, and noticethat after you have been traveling for 1 hour ( ,t= 1), you pass a sign saying it is 110miles to Seattle, and after driving another half-hour you pass a sign saying it is 85 milesto Seattle. Using the horizontal axis for the timetand the vertical axis for the distanceyfrom Seattle, graph and find the equationy=mt+bfor your distance from Seattle. Findthe slope,y-intercept, andt-intercept, and describe the practical meaning of graph ofyversustis a straight line because you are traveling at constant line passes through the two points (1,110) and ( ,85), so its slope ism= (85 Lines17110)/( 1) = 50.

8 The meaning of the slope is that you are traveling at 50 mph;misnegative because you are travelingtowardSeattle, , your distanceyisdecreasing. Theword velocity is often used form= 50, when we want to indicate direction, while theword speed refers to the magnitude (absolute value) of velocity, which is 50 mph. Tofind the equation of the line, we use the point-slope formula:y 110t 1= 50,so thaty= 50(t 1) + 110 = 50t+ meaning of they-intercept 160 is that whent= 0 (when you started the trip) you were160 miles from Seattle. To find thet-intercept, set 0 = 50t+160, so thatt = 160/50 = meaning of thet-intercept is the duration of your trip, from the start untilyou arrivein Seattle. After traveling 3 hours and 12 minutes, your distanceyfrom Seattle will be the equation of the line through (1,1) and ( 5, 3) in the formy=mx+b. the equation of the line through ( 1,2) with slope 2 in the formy=mx+b.

9 The equation of the line through ( 1,1) and (5, 3) in the formy=mx+b. the equationy 2x= 2 to the formy=mx+b, graph the line, and find they-intercept andx-intercept. the equationx+y= 6 to the formy=mx+b, graph the line, and find they-interceptandx-intercept. the equationx= 2y 1 to the formy=mx+b, graph the line, and find they-intercept andx-intercept. the equation 3 = 2yto the formy=mx+b, graph the line, and find they-interceptandx-intercept. the equation 2x+ 3y+ 6 = 0 to the formy=mx+b, graph the line, and find they-intercept andx-intercept. whether the lines 3x+ 6y= 7 and 2x+ 4y= 5 are parallel. a triangle in thex, y plane has vertices ( 1,0), (1,0) and (0,2). Find the equationsof the three lines that lie along the sides of the triangle iny=mx+bform. that you are driving to Seattle at constant speed. After you have been travelingfor an hour you pass a sign saying it is 130 miles to Seattle, and after driving another 20minutes you pass a sign saying it is 105 miles to Seattle.

10 Using the horizontal axis for thetimetand the vertical axis for the distanceyfrom your starting point, graph and find theequationy=mt+bfor your distance from your starting point. How long does the trip toSeattle take? for temperature in degrees Celsius (centigrade), and letystand for temperature indegrees Fahrenheit. A temperature of 0 C corresponds to 32 F, and a temperature of 100 Ccorresponds to 212 F. Find the equation of the line that relates temperature Fahrenheitytotemperature Celsiusxin the formy=mx+b. Graph the line, and find the point at whichthis line intersectsy=x. What is the practical meaning of this point? 18 Chapter 1 Analytic car rental firm has the following charges for a certain type of car: $25 per day with 100free miles included, $ per mile for more than 100 miles. Suppose you want to rent acar for one day, and you know you ll use it for more than 100 miles.