Section 9.2: Lines of Fit

Property

A trend line (or regression line) models the relationship between two variables, coming as close as possible to all data points. To find its equation, pick two points on the drawn line, which do not need to be original data points.

• First, calculate the slope ($m$) using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, which represents the vertical change divided by the horizontal change.
• Next, identify the y-intercept ($b$), which is the point where the line crosses the y-axis and occurs when $x$ is zero.
• Finally, substitute $m$ and $b$ into the slope-intercept form, $y = mx + b$.

Examples

• Find the slope between $(2, 3)$ and $(7, 9)$ using the formula: $m = \frac{9 - 3}{7 - 2} = \frac{6}{5}$.
• A regression line passes through $(5, 1.25)$ and $(25, 3.35)$. The slope is $m = \frac{3.35 - 1.25}{25 - 5} = 0.105$. The equation simplifies to $y = 0.105x + 0.725$.
• If the calculated slope of a line of fit is $m = \frac{2}{3}$ and the y-intercept is $b = -4$, the complete equation is $y = \frac{2}{3}x - 4$.

Explanation

The regression line is a straight line that best summarizes the trend in a scatterplot. The slope formula is a way to calculate the steepness of this line without relying solely on a visual graph. Once you calculate the slope and identify the y-intercept, substituting your specific numerical values into $y = mx + b$ gives you a final linear model that represents the overall trend of the data.

Property

The slope-intercept form for a linear equation is $y = mx + b$, where $m$ is the slope of the line and the point $(0, b)$ is the y-intercept.

The slope formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $m = \frac{y_2 - y_1}{x_2 - x_1}$.

This formula calculates the ratio of the change in the y-coordinates (rise) to the change in the x-coordinates (run).