Study Notes on Lines and Their Equations

Graphs, Functions, and Models

Lines

Lines are one of the most fundamental objects in precalculus, forming the basis for understanding more complex functions and their graphs. This section covers the definition, equations, and properties of lines in the coordinate plane.

• Definition: A line is the set of all points in the plane that satisfy a linear equation of the form $y = mx + b$, where m is the slope and b is the y-intercept.

• Slope (m): The slope of a line measures its steepness and is calculated as the ratio of the vertical change to the horizontal change between two points on the line.

Formula for Slope:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

• y-intercept (b): The y-intercept is the value of y where the line crosses the y-axis (when $x = 0$).

• Equation of a Line: The most common forms are:

  • Slope-Intercept Form: $y = mx + b$

  • Point-Slope Form: $y - y_1 = m(x - x_1)$

  • Standard Form: $Ax + By = C$

Example: Finding the Equation of a Line

Given two points, $(x_1, y_1)$ and $(x_2, y_2)$, the equation of the line passing through them can be found as follows:

1. Calculate the slope $m$ using the formula above.

2. Use the point-slope form with one of the points.

3. Simplify to the desired form (usually slope-intercept or standard form).

Example: Find the equation of the line passing through $(2, 4)$ and $(7, 12)$.

$m = \frac{12 - 4}{7 - 2} = \frac{8}{5}$ Using point-slope form with $(2, 4)$: $y - 4 = \frac{8}{5}(x - 2)$

Simplifying to slope-intercept form:

$y = \frac{8}{5}x + \left(4 - \frac{16}{5}\right) = \frac{8}{5}x - \frac{16}{5} + 4 = \frac{8}{5}x + \frac{4}{5}$

Graphing Lines

• To graph a line, plot the y-intercept, then use the slope to find another point.

• Draw a straight line through the points.

Special Cases

• Horizontal lines: $y = c$ (slope $m = 0$)

• Vertical lines: $x = c$ (undefined slope)

Example: Horizontal and Vertical Lines

• Horizontal: $y = 3$

• Vertical: $x = -2$

Additional info:

• Lines are the simplest polynomial functions (degree 1).

• Understanding lines is essential for studying more complex functions and their transformations.