Section 13 Graphing Data

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Figure 1-15 The independent variable, mass, is on the horizontal axis The graph shows that the length of the spring increases as the mass suspended from the spring increases

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Plotting Line Graphs

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Use the following steps to plot line graphs from data tables 1 Identify the independent and dependent variables in your data The independent variable is plotted on the horizontal axis, the x-axis The dependent variable is plotted on the vertical axis, the y-axis

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Math Handbook

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Graphs of Relations pages 848 852

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2 Determine the range of the independent variable to be plotted 3 Decide whether the origin (0, 0) is a valid data point 4 Spread the data out as much as possible Let each division on the graph paper stand for a convenient unit This usually means units that are multiples of 2, 5, or 10 5 Number and label the horizontal axis The label should include the units, such as Mass (grams) 6 Repeat steps 2 5 for the dependent variable 7 Plot the data points on the graph 8 Draw the best-fit straight line or smooth curve that passes through as many data points as possible This is sometimes called eyeballing Do not use a series of straight line segments that connect the dots The line that looks like the best fit to you may not be exactly the same as someone else s There is a formal procedure, which many graphing calculators use, called the least-squares technique, that produces a unique best-fit line, but that is beyond the scope of this textbook 9 Give the graph a title that clearly tells what the graph represents

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Figure 1-16 To find an equation of the line of best fit for a linear relationship, find the slope and y-intercept

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Linear Relationships

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Scatter plots of data may take many different shapes, suggesting different relationships (The line of best fit may be called a curve of best fit for nonlinear graphs) Three of the most common relationships will be shown in this section You probably are familiar with them from math class When the line of best fit is a straight line, as in Figure 1-15, the dependent variable varies linearly with the independent variable There is a linear relationship between the two variables The relationship can be written as an equation Linear Relationship Between Two Variables y mx b

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Length of a Spring for Different Masses 170 160 Length (cm)

rise run

140 b 0 5 137

10 15 20 25 30 35 Mass (g)

Find the y-intercept, b, and the slope, m, as illustrated in Figure 1-16 Use points on the line they may or may not be data points

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The slope is the ratio of the vertical change to the horizontal change To find the slope, select two points, A and B, far apart on the line The vertical change, or rise, y, is the difference between the vertical values of A and B The horizontal change, or run, x, is the difference between the horizontal values of A and B

y rise run x The slope of a line is equal to the rise divided by the run, which also can be expressed as the change in y divided by the change in x

Slope

In Figure 1-16: m

(160 cm (30 g

141 cm) 5 g)

008 cm/g If y gets smaller as x gets larger, then y/ x is negative, and the line slopes downward The y-intercept, b, is the point at which the line crosses the y-axis, and it is the y-value when the value of x is zero In this example, b 137 cm When b 0, or y mx, the quantity y is said to vary directly with x

Figure 1-17 This graph indicates a quadratic, or parabolic, relationship

Nonlinear Relationships

Figure 1-17 shows the distance a brass ball falls versus time Note that the graph is not a straight line, meaning the relationship is not linear There are many types of nonlinear relationships in science Two of the most common are the quadratic and inverse relationships The graph in Figure 1-17 is a quadratic relationship, represented by the following equation Quadratic Relationship Between Two Variables y ax2 bx c