Linear Functions

A linear function is a function that can be graphically represented as a straight line. It has the form $$y = mx + b$$ , where $$m$$ is the slope and $$b$$ is the y-intercept.

The slope of a line represents the rate of change of the dependent variable (y) with respect to the independent variable (x). It indicates how steep the line is.

The slope (m) can be calculated using the formula: $$m = \frac{{y2 - y1}}{{x2 - x1}}$$ .

The slope-intercept form of a linear equation is $$y = mx + b$$ , where $$m$$ is the slope and $$b$$ is the y-intercept.

The y-intercept is the point where the line crosses the y-axis. It is represented by the value of $$b$$ in the slope-intercept form $$y = mx + b$$ .

An undefined slope indicates a vertical line, where the change in x is zero, leading to division by zero in the slope formula.

A slope of zero indicates a horizontal line, meaning there is no change in y as x changes.

To convert from standard form $$Ax + By = C$$ to slope-intercept form, solve for y to isolate it: $$y = -\frac{A}{B}x + \frac{C}{B}$$ .

The slope can represent various rates of change in real-world contexts, such as speed, cost per item, or growth rates.

Use the point-slope form: $$y - y1 = m(x - x1)$$ , where $$(x1, y1)$$ is the point and $$m$$ is the slope.