Linear Functions

A linear function is a function that represents a straight line on a graph. Its general form is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. Linear functions are fundamental in mathematics and are used to model relationships where one quantity changes at a constant rate with respect to another.

Slope

The slope ($m$) measures the steepness and direction of a line. It is the ratio of the change in the vertical axis (rise) to the change in the horizontal axis (run) between any two points on the line. A positive slope indicates the line is increasing from left to right, while a negative slope indicates the line is decreasing. The formula for slope is given by:

(1, 2)(3, 6)runrise
$$ m = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} $$

For a line passing through points $(1, 2)$ and $(3, 6)$, the slope is: $$ m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2 $$

Y-Intercept

The y-intercept ($b$) is the point where the line crosses the vertical (y) axis. At this point, the value of $x$ is always zero, so the coordinates of the y-intercept are $(0, b)$. It represents the initial value of the function when the independent variable is zero.

(0, 3)

In the function $y = 2x + 3$, the y-intercept is $b = 3$. This means the line crosses the y-axis at the point $(0, 3)$.