Math Handbook - Glencoe

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APPENDIX B Math Handbook a Linear curve y x b Inverse curve y 1 x Figure 15 The shape of the curve formed by a plot of experimental data indicates how the variables are related. c Exponential curve y x n (n 1) d Root curve n y x (n 1) Direct and inverse relationships In your study of chemistry, the most common curves are the linear, representing the direct relationship (y x), and the inverse, representing the inverse relationship (y 1/x), where x represents the independent variable and y represents the dependent variable. In a direct relationship, y increases in value as x increases in value or y decreases when x decreases. In an inverse relationship, y decreases in value as x increases in value. An example of a typical direct relationship is the increase in volume of a gas with increasing temperature. When the gases inside a hot air balloon are heated, the balloon gets larger. As the balloon cools, its size decreases. However, a plot of the decrease in pressure as the volume of a gas increases yields a typical inverse curve. You also may encounter exponential and root curves in your study of chemistry. See Figure 15cand d. What types of relationships between the independent and dependent variables do the curves describe? How do the curves differ from a and b?An exponential curve describes a relationship in which one variable is expressed by an exponent. And, a root curve describes a relationship in which one variable is expressed by a root. What type of relationship is described by the curve in Figure 16? The linear graph The linear graph is useful in analyzing data because a linear relationship can be translated easily into equation form using the equation for a straight line y mx b Strontium-90 remaining (%) 100 50 25 12.5 10.0 g Figure 16 5.00g 2.50 g 1.25g 0 0 1 2 3 4 Number of half-lives (1 half-life = 29 years) Half-life is the amount of time it takes for half of a sample of a radioactive isotope to decay (Chapters 4 and 25). Notice that as the number of half-lives increases, the amount of sample decreases. Math Handbook where y stands for the dependent variable, m is the slope of the line, x stands for the independent variable, and b is the y-intercept, the point where the curve crosses the y-axis. Math Handbook 905
APPENDIX B Math Handbook Figure 14 a To plot a point on a graph, place a dot at the location for each ordered pair (x,y) determined by your data. In this example, the dot marks the ordered pair (40 mL, 40 g). b Generally, the line or curve that you draw will not include all of your experimental data points. Mass (g) 70 60 50 40 30 20 Density of Water Mass (g) 70 60 50 40 30 20 Experimental Data 10 10 0 0 10 20 30 40 50 60 70 Volume (mL) a 0 0 10 20 30 40 50 60 70 Volume (mL) b Math Handbook After the scales are chosen, plot the data. To graph or plot an ordered pair means to place a dot at the point that corresponds to the values in the ordered pair. The x-coordinate indicates how many units to move right (if the number is positive) or left (if the number is negative). The y-coordinate indicates how many units to move up or down. Which direction is positive on the y-axis? Negative? Locate each pair of x- and y-coordinates by placing a dot as shown in Figure 14a. Sometimes, a pair of rulers, one extending from the x-axis and the other from the y-axis, can ensure that data is plotted correctly. Drawing a curve Once the data is plotted, a straight line or a curve is drawn. It is not necessary to make it go through every point plotted, or even any of the points, as shown in Figure 14b. Graphing of experimental data is an averaging process. If the points do not fall along a line, the best-fit line or most probable smooth curve through the points should be drawn. When drawing the curve, do not assume it will go through the origin (0,0). Naming a graph Last but not least, give each graph a title that describes what is being graphed. The title should be placed at the top of the page, or in a box on a clear area of the graph. It should not cross the data curve. Once the data from an experiment has been collected and plotted, the graph must be interpreted. Much can be learned about the relationship between the independent and dependent variables by examining the shape and slope of the curve. Using Line Graphs The curve on a graph gives a great deal of information about the relationship between the variables. Four common types of curves are shown in Figure 15. Each type of curve corresponds to a mathematical relationship between the independent and dependent variables. 904 Chemistry: Matter and Change
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Math Handbook - Glencoe

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