The Rectangular Coordinate System
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1 The Mathematics Competenc Test The Rectangular Coordinate Sstem When we write down a formula for some quantit,, in terms of another quantit,, we are epressing a relationship between the two quantities. For eample, if we use the smbols and = the area of a square = the length of one side of the square, then the formula = tells us the relationship between the length of a side of a square and the area of the square. It tells us that to calculate the area of the square, we must raise the length of one side of that square to the power (which is one reason wh raising a number to the power is often referred to as squaring the number). We can use this formula to calculate the value of for an particular value of. This formula relating to is itself informative, but often we can understand the nature of the relationship between and even better if we have a visual image of its characteristics as well. This is where graphing formulas is helpful. A graph of a relationship is a wa of drawing points and other geometric shapes at locations representing the values of and. This is most commonl done using a so-called rectangular coordinate sstem. When the formula epresses in terms of, the coordinate sstem is usuall arranged as: the origin ( =, = ) horizontal ais or - ais vertical ais or - ais positive values of positive values of negative values of negative values of the vertical ais and the horizontal ais (often called the -ais and the -ais, respectivel, if the two variables are and ) intersect at a central point called the origin, which corresponds to = and =. David W. Sabo () The Rectangular Coordinate Sstem Page of 6
2 a numerical scale is created on each ais. Values on the horizontal ais increase from at the origin though positive values to the right, and from at the origin through negative values: -, -,,, etc., to the left. The scales along the aes should be uniform the values of should be spaced uniforml along the length available. values on the vertical scale increase from at the origin through positive values as ou go upwards, and from at the origin through negative values: -, -,,, etc., as ou go down. These two aes complete with the eplicitl labelled numerical scales form what is called a rectangular coordinate sstem. Then, the corresponding pair of values, = a and = b, written as a pair of numbers in this order in brackets, (a, b), corresponds to, or is plotted as a point at the location where the vertical line through = a intersects the horizontal line through = b: = b the point (a, b) a units The values = a and = b here are called the coordinates or rectangular coordinates of the point. You can also think of the coordinates, (a, b), of a point as indicating that to get to the point from the origin, ou need to move a units horizontall and b units verticall. Positive movements are to the right horizontall, and upwards verticall. Negative movements are to the left horizontall and downwards verticall. Eample: Plot the points and A = (, ) B = (, ) C = (-, ) D = (, -) on the coordinate aes shown to the right. Be sure to label the aes scales and label the points ou plot. Solution: Recalling the meaning of this notation giving pairs of numbers in brackets, we know that the point A is the point that occurs at = and = the first number in the brackets gives the - coordinate of the point, and the second number in brackets gives the -coordinate of the point. David W. Sabo () The Rectangular Coordinate Sstem Page of 6
3 Now, the -coordinates for these four points range from a minimum of - to a maimum of, so our horizontal scales must go at least to - on the left to at least + on the right. Also, we see that the -coordinates must go at least to on the bottom to at least on the top. The result is: B = (, ) A = (, ) C = (-, ) D = (, -) The dotted lines show how the points line up with the appropriate scale positions on both aes. Notice that the points are plotted as heav dots if ou are just plotting points, there is no need to join them b lines or add an other features to the graph. Remarks: (i) When the coordinates of a point are written as a bracketed pair of numbers, we alwas write the horizontal coordinate first and the vertical coordinate second. This is wh (a, b) is often called an ordered pair. For eample, the point (, ) is at a quite different location than the point (, ): (, ) (, ) (ii) Since the identit of a point comes from its measured position or location with respect to the scales on the coordinate aes, it is mandator to show the scales on the aes eplicitl, and to label the aes eplicitl. If ou omit the scale markings and labels, ou end up with a meaningless graph. (iii) The graph of a single point is just a dot at the appropriate location. No additional lines, etc. should be drawn unless the are requested. David W. Sabo () The Rectangular Coordinate Sstem Page of 6
4 (iv) Points on the horizontal ais have coordinates of the form (b, ) that is, their vertical coordinates are equal to zero. Points on the vertical ais have coordinates of the form (, b) that is, their horizontal coordinate is zero. The origin has coordinates (, ) both of its coordinates are zero. (-, ) (, ) (, ) - - (, ) (, ) (v) The rectangular coordinate aes divide the plane into four regions, called quadrants. The quadrants are identified b number, with quadrant, or the first quadrant being the upper right one. The are arranged as shown: second quadrant first quadrant third quadrant fourth quadrant Plotting Graphs of Formulas To graph a formula b hand, the usual procedure is to i) make a table of and values for a representative collection of values of in the specified interval ii) plot these pairs of values as points on a rectangular coordinate sstem iii) join the points b a line or smooth curve Eample: Plot the graph of = + for between and + inclusive. Solution: (i) start b making a table of coordinates of representative points: David W. Sabo () The Rectangular Coordinate Sstem Page of 6
5 [ = () + = ] - - [ = (-) + = - ] - [ = (-) + = ] etc. 7 9 (ii) we need coordinate aes that have going from to + and going from to +9. Set these up and then plot the points as dots at the appropriate locations: (-, ) - (-, -) (, ) (, 7) (, ) (, ) (, 9) (iii) The points appear to lie on a straight line. Laing a straight edge on the graph confirms this, so in this case, just draw a straight line through the points to complete the graph, as shown above. Had we recognized that the formula, = +, is the tpe of formula that gives a straight line graph, we could have saved ourselves some work, since we would need to plot onl two points to get the entire graph. Eample: The area, A, of a square with sides of length s is given b the formula A = s Plot a graph of A vs s for s = through s =. Solution: This eample illustrates several issues: (i) People often use the smbols and genericall when speaking about graphs. However, we can create a rectangular coordinate sstem for an pair of smbols or variables we wish to use. (ii) When asked to plot the graph of A vs. s, we are to make the vertical ais the A-ais, and the horizontal ais is the s-ais: David W. Sabo () The Rectangular Coordinate Sstem Page of 6
6 A s So, here, we start again b making a table of coordinates of representative points for the graph: s A = s 9 6 Now, plot the points on a rectangular coordinate sstem which has a horizontal (s) ais running from s = to s =, at least, and a vertical (A) ais running at least from A = to A =. A (, ) (, 6) (, ) (, 9) (, ) (, ) s We see in this case that the points appear to follow a curved path, bending upwards as ou move towards the right. The graph starts out with quite a shallow slope around s =, but appears to get steeper and steeper as we move towards higher values of s. David W. Sabo () The Rectangular Coordinate Sstem Page 6 of 6
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