Physics Study Guide | Linear Relationships
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.
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.
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, Δy, is the difference between the vertical values of A and B. The horizontal change, or run, Δx, is the difference between the horizontal values of A and B.
Slope $m = \dfrac{\text{rise}}{\text{run}} = \dfrac{\Delta y}{\Delta x}$
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.
In Figure 1-16: $m = \dfrac{(16.0\ \text{cm} - 14.1\ \text{cm})}{(30\ \text{g} - 5\ \text{g})} = 0.08\ \text{cm/g}$
If y gets smaller as x gets larger, then Δy/Δx 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 = 13.7 cm. When b = 0, or y = mx, the quantity y is said to vary directly with x.