Graphing Linear Equation, Linear Function (First Degree Polynomial)

Lines parallel to the axes, horizontal and vertical lines
The point-slope form of a line

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
The point-slope form of a line
The equation of the line y = mx + c is defined by the slope m and by one of its points P1(x1, y1).
As coordinates of the point P1 must satisfy the equation of the line, we calculate the y-intercept, by substituting them into the equation of the line,
P1(x1, y1)    =>       y = mx + c
y1 = mx1 + c    or       c = y1 - mx1
Therefore, the line that passes through the given point P1(x1, y1) and given is its slope m, has the equation
 y = mx + y1 - mx1 or y - y1 = m(x - x1)
Example:  Find the equation of the line that is parallel with the line  y = - x - 2 and passes through the point P1( 2, 1) .
Parallel and perpendicular lines
Two lines having slopes m1 and m2 are parallel if
 m1 = m2 that is, if they have the same slope.
To acquire the criteria when two lines,  y = m1x and  y = m2are perpendicular or orthogonal we can use the principle of similar triangles, OA'A and OB'B in the picture.
 Therefore, m1 : 1 = -1 : m2 =>

This relation will stay unchanged if we translate the perpendicular lines, that is, when lines
y = m1x + c1 and  y = m2x + care written in the slope-intercept form.
Two lines are perpendicular if the slope of one line is the negative reciprocal of the other.
 Example:  Find the equation of the line that is perpendicular to the line and passes through
the point A(-2, 5).
Beginning Algebra Contents C