Linear Forms

 

  1. slope - intercept form: y = mx +b, where m is the slope and b is the y-intercept

              example:  y = 5x -9, where the slope is 5 and intercept is -9

  1. point - slope form: y – y1 = m(x-x1), where the point is denoted by (x1, y1)

example: y – 5 = 2(x +8), where the slope is 2 and the point is (-8, 5)

  1. horizontal form: y = b (line is parallel to the x-axis)

example: y = 10 or y = -6

  1. vertical form: x = a (line is parallel to the y-axis)

example:  x = 4 or x = -23

  1. general form:  ax + by +c = 0

example: 8x – 7.3y +9.245 = 0 or 8x -7.3y = – 9.245

 

Slope

          The slope of a line is generally considered the “steepness” of the line; it is often seen as the ratio:                                                                         rise   

                                               run

Another way to think about this is the change in the y distance over the change in the x distance between two points.

The formula for the slope is the following:

 

m = y2 – y1        Notice that the difference in the y distance is in the numerator and

        x2 – x1    and the difference in the x distance is in the denominator AND the

                      difference is found in the SAME direction in both.

 

Find the slope of the line passing through (4, 2) and (90, 12).       (m = 10/86 or 0.11)

 

The slopes used in models are reported in decimal form, usually two places is sufficient.

 

Slope-intercept Form

 

Find the slope-intercept form of the line above in gray.

 

Y = mx + b                      So, the slope-intercept form of the equation is:

Y = 0.11x + b                         y = 0.11x + 1.56

2 = 0.11(4) + b

2 = 0.44 + b

1.56 = b

 

Point-Slope Form

 

Find the point-slope equation of the line in gray above.

 

Y – y1 = m(x – x1)   y -2 = 0.11(x -4)  or   y – 12 = 0.11(x – 90)