Graphing Linear Equation, Linear Function (First Degree Polynomial)

Linear function f (x) = mx + c

Linear function  f (x) = mx + c
The expression
y = mx + c    or      f (x) = ax + b
we call linear function, where x is the argument or the independent variable,  f (x) or y is the dependent variable or the function, numbers,  a and b or  m and c are constants.
If the coefficient m = 0, then y does not depend of x.
Linear function  f (x) = mx + cm is not 0, to each value of the argument x associates a unique value of y.
The graph of the linear function
A function y = f (x) can be considered as the set of ordered pairs (x, y) where each pair represents a point in a Cartesian coordinate system xOy.
When plotted on an x, y graph linear function forms a straight line.
The constant m is called the slope or gradient and the constant c is the y-intercept. This is the point of intersection between the graph of the function and the y-axis.
If the coefficient m is not 0 and c = 0 then, the linear function has the form  y = f (x) = mx.

To the graph of the function f (x) = mx belong the points P (x, mx ) of a coordinate plane of which the abscissas are values of argument x taken arbitrarily, and the ordinates are the calculated values of the function.

Thus, for x = 0  =>  f (x) = f (0) = m · 0 = 0, this means that the graph of the function f (x) =  mx is running through the origin O(0, 0) or the x-intercept is at the origin.
As the above right triangles OAP1 and OBP are similar the ratios of the ordinates and the abscissas of the points P and P1 are equal:
 or
 The value of the ratio
determines the rate at which the y-coordinate of a straight line changes with respect to the x-coordinate.
On a line graph, changing m makes the line steeper or more gentle (shallower).
If m is negative, y decreases as x increases.
Example:  Find the equation of the line that passes through the origin and the point A(-3, 2).
By plugging the coordinates of the point A into the equation y =  mx we determine the slope m.
Beginning Algebra Contents C