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Linear functions are those whose graph is a straight line. A linear function has one independent variable and one dependent variable. The independent variable is x and the dependent variable is y. a is the constant term or the y intercept.
Inspect the graph to see if any vertical line drawn would intersect the curve more than once. If there is any such line, the graph does not represent a function. If no vertical line can intersect the curve more than once, the graph does represent a function.
In Bishop’s book of Pattern Recognition & Machine Learning, there are a few examples where the fit is a curve or a straight line. The term linear means that the fit should be a linear function or a polynomial of degree 1 i.e., a straight line.
Linear functions graph as a straight line, no curves allowed. So, if the graph is a straight line, it is the graph of a linear function. From a table, you can verify a linear function by examining the x and y values. The rate of change for y with respect to x remains constant for a linear function.
So far, all of the graphs discussed have had a line that either always goes up or always goes down (the technical term for this is monotonic). Other relationships are more complex, with lines than can go up and down (called non-monotonic).
A linear relationship (or linear association) is a statistical term used to describe a straight-line relationship between two variables. Linear relationships can be expressed either in a graphical format or as a mathematical equation of the form y = mx + b.
What the Slope Means. A positive slope means that two variables are positively related—that is, when x increases, so does y, and when x decreases, y also decreases. Graphically, a positive slope means that as a line on the line graph moves from left to right, the line rises.