Modeling 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 linear function represents the rate of change of the function. It indicates how much the dependent variable (y) changes for a unit change in the independent variable (x).

The y-intercept is the point where the line crosses the y-axis. It is the value of $$y$$ when $$x = 0$$ .

A positive linear correlation indicates that as one variable increases, the other variable also increases. The graph of this relationship slopes upwards.

A negative linear correlation indicates that as one variable increases, the other variable decreases. The graph of this relationship slopes downwards.

The formula for a linear model is typically expressed as $$y = mx + b$$ , where $$m$$ is the slope and $$b$$ is the y-intercept.

A line is a good estimate of best fit if it minimizes the distance between the line and all the data points, often assessed using the least squares method.

Correlation indicates that two variables are related, while causation indicates that one variable directly affects the other.

The correlation coefficient quantifies the strength and direction of a linear relationship between two variables, ranging from -1 to 1.

To write a linear equation from a word problem, identify the variables, determine the slope and y-intercept from the context, and express it in the form $$y = mx + b$$ .