Graphing Linear Equation, Linear Function (First Degree Polynomial)
      Linear function  f (x) = mx + c
         Slope-intercept form of a line
         Slope or gradient, y-intercept and x-intercept or zero of a function
         The graph of the linear function
      Properties of the linear function
      Lines parallel to the axes, horizontal and vertical lines
Linear function  f (x) = mx + c
The expression,  y = mx + c or  y = a1x + a0,  we call linear function or slope-intercept form of the equation of the line. Where, the constant m is called the slope or gradient and the constant c is the y-intercept.
The slope of a line is the ratio of its vertical change to its horizontal change, or it is the tangent of the angle between the direction of the line and the x-axis. 
The y-intercept is the point of intersection between the graph of the function and the y-axis.
The zero of a function or the x-intercept is the value of the independent variable x at which the value of the function is zero.
The graph of the linear function
 
this expression, in a coordinate system, represents the translation of the source linear function  y = mx  in the direction of the y-axis by  y0 = c or the translation in the direction of the x-axis by  x0 = - c/m  as is shown in the figure above.
Therefore, the linear expression we also write as 
y = a1x + a0   or   y = a1(x - x0)   or   y - y0 = a1x, where  y0 = a0
To find the zero or the x-intercept of the linear function set  y = 0  and solve the equation for x, i.e.,
                                       y = 0    =>    0 = a1x + a0
Example:   Find the equation of the line that passes through the origin and the point A(-3, 2). Translate the line in the direction of the x-axis by x0 = - 3, then find its equation.
Calculate the slope m by plugging the coordinates of the point A into the equation  y =  mx,
   
By plugging x0 = - 3 into  y = a1(x - x0)
Properties of the linear function
We examine the behavior of a function y = f (x) by moving from left to right in the direction of x-axis by inspecting its graph.
The linear function  f (x) = mx + c, m > 0 is increasing, the graph rises from left to right, that is, 
                   f(x1) <  f(x2)   for all  x1 < x2
If m < 0 linear function decreasing,   f (x1) >  f (x2)   for all  x1 < x2    i.e., the graph falls from left to right.
Lines parallel to the axes, horizontal and vertical lines
If m = 0, the function does not depend of x. To every number x associated is the same constant value y = c.
A line parallel to the x-axis is called a horizontal line (or constant).
If the x value never changes a line is parallel to the y-axis. A line parallel to the y-axis is called a vertical line.
The linear function changes the sign at the root or zero point. 
Thus, if m > 0, then f (x< 0 for all   while f (x) > 0 for all  
   That is,   f (x) = mx + c, m > 0 is negative for all x less than the root, positive for all x greater than the root, and at the root  f (x) = 0.
If  m < 0, then f (x> 0 for all   while f (x) < 0 for all  
Intermediate algebra contents
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