Graphs are of two general kinds: those representing equations or formulas, and those depicting observed data, such as statistical or experimental results. In either case, the graph is a pictorial way of showing arithmetic or algebraic relations between numbers. :RA Graphs of Equations. There are many sets of values of the letters that satisfy an equation. The equation y = 2x - 3 is satisfied when x = 2 and y = 1, when x = 0 and y = -3, and so on. By convention, we write such number-pairs as (2,1) and as (0,-3). Always put the x value (or what corresponds to x if there are different letters) before the y value. These number pairs are algebraic in character. :RA To each of the number pairs that satisfy an equation there corresponds a point on the graph of that equation. When you plot these points and join them, you have the graph of the equation. Then to draw the graph of the equation 3x - y = 2, first solve for y: y = 3x - 2. Then make a table of number pairs. x y ------ 0 -2 1 1 2 4 :RA :SD :SB :SG217104202251023143 :SP245024210144210144210144 :SH0636(3,7) :SH0935(2,4) :SH1234(1,1) :SH1533(0,-2) :SF The number pairs are plotted as shown in the accompanying graph, and the points are joined by a straight line through them. Note that the line does not stop at the points plotted but continues through them to show that there are many other points (number pairs) which satisfy the equation. :RA :SB :SG217104202251023143 :SP245024210144210144210144 :SH0636(3,7) :SH0935(2,4) :SH1234(1,1) :SH1533(0,-2) :SF Point (3,7) is on the graph. You can see that x = 3, y = 7 will satisfy the equation 3x - y = 2. To each of the points on a graph, there corresponds a pair of numbers which fulfill the conditions of the question. :RA :SD When you have an equation in which x and y appear only in the first power, and no term contains their product, the graph is a straight line and the equation is said to be linear. For such graphs only three points are needed. :RA Equations such as: xy = 6, 2 2 x + y = 9, 2 or y = x - 2x are not linear. Their graphs are curves instead of lines. :RA :SD :SB :SG210128195237031143 :SP231040210088210088210088 :SH0734(2,9) :SH0933(1,7) :SH1132(0,5) :SH0431c :SH1635n :SF Formulas are equations. Note: the graph of the formula c = 2n + 5 is drawn in the same way. Note in the graph that n corresponds to x, so that it becomes the horizontal axis. Also, when a formula has some physical significance, negative values may have no meaning. n c ------ 0 5 1 7 2 9 :RA :SD :SB :SG224096195258031127 :SP210040245120238104238080 :SF Consider the graph shown. To find y when x = 2, you go two divisions on the x axis to the right of the origin (0,0); then draw a vertical line. This may be done mentally. Where this line intersects the graph, find the y value of the point. y = -1, is the Ans. :RA :SD To find the equation of the line, recall that two points fix (determine) a straight line. Every linear equation can be solved for y and put in the form; y = mx + b. Note that b is the value of y when x = 0. Now find two points on the line: (0,3) and (2,-1). Since these two number pairs must satisfy the equation, 3 = m(0) + b and -1 = m(2) + b. From this we derive b = 3 and m = -2. The equation of the line is then y = -2x + 3 or 2x + y = 3. :RA Note that if a line goes through the origin, point (0,0), the value of b is 0 and the equation becomes of the form y = mx. :RA Here are several general equations to solve common graph problems. Formula to find the distance between (x,y) and (x',y'). ____________________ / 2 2 d = \/(x - x') + (y - y') Formula to find the midpoint between (x,y) and (x',y'). x + x' y + y' m = ------ , ------ 2 2 :RA Formula to find the slope between (x,y) and (x',y'). y' - y m = ------ x' - x :RA :SD :SB :SC192035200054226036226036 :SP215073200054243060243060 :SH0631a :SH0729b :SH0932x :SH0732c :SH1324a = 90^ 25% :SH1424b = 144^ 40% :SH1524c = 60^ 16 2/3% :SF Graphs of Statistical Data The Circle Graph. This type of graph is used to show the component parts of a whole. The circle is divided into sectors by radii. Since a sector with a central angle of 90^, is 90/360 or 1/4 of the circle, such a sector represents 25%, provided the circle corresponds to the full 100%. :RA :SB :SC192035200054226036226036 :SP215073200054243060243060 :SH0631a :SH0729b :SH0932x :SH0732c :SH1324a = 90^ 25% :SH1424b = 144^ 40% :SH1524c = 60^ 16 2/3% :SF A 60^ sector shows 1/6 or 16 2/3%. Any sector is then the same fraction (or percent) of the circle as its central angle is of 360^. In the circle, angle x is found by subtracting the sum of the other angles from 360^. Angle x = 66^, and its sector shows 18 1/3%. :RA :SD The Bar Graph. This type consists of bars (horizontal or vertical). Only the length of the bars is measured to scale. The thickness has no meaning, although for the sake of appearance, we usually keep the bars of the same thickness. :RA The Broken-Line Graph. On this type of graph, points are plotted in the same way as points on a linear equation graph. There are, however, several differences between the two types of graphs. :RA First, on the broken-line graph, there is no set rule; rather, the points are found from given values. Second, consecutive points are joined by straight lines which may differ in direction. Third, the graph is shown only for the points given, since there is no rule to determine a point. Fourth, either or both of the scales may not start at zero. In fact, they may be different, so that care must be exercised in reading horizontal and vertical values. :RA Note: When the changes to be represented on the graph are continuous (as growth in weight or height, or daily temperatures), a broken-line graph is sometimes "smoothed out" into a curved-line graph. :ET :ET ' !m.. ' !DISK1 j!nBASICA EXE `qåÇ√EDLIN COM `qÜá SAMPLE BAK ¿Γ2|WDISK2 m!oDISK3 o!pDISK4 p!rDISK5 r!sDISK6 t!tDISK7 u!uDISK8 w!vDISK9 x!wDISK10 z!xDISK11 }!y├^ ¼ V ó$ë "╟╬╟║45╟
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