Point-Slope Form

Slope-intercept form is useful when we know the y- intercept of a line. However, we are not always given this information. When we know the slope and one point which is not the y-intercept, we can write the equation in point-slope form.
Equations in point-slope form look like this:

m= y-y1
x-x1

And can be rewritten as:

y-y1=m(x-x1)

Where m is the slope of the line and (y1,x1) are points on the line (any points on the the line will work).
To write an equation in point-slope form, given a graph of that equation, first determine the slope by picking two points. Then pick any point on the line and write it as an ordered pair (y1,x1). It does not matter which points you pick, as long as they are on the line different points yield different constants, but the resulting equations will describe the same line.
Finally, write the equation, substituting numerical values in for m, x, and y. Check your equation by picking a point on the line, not the point you chose as (y1,x1), and confirming that these points satisfy the equation.
So that:
(y1,x1)

In other words:

The slope equation says that the slope of a line is found by determining the amount of rise of the line between any two points divided by the amount of run of the line between the same two points. A method for carrying out the calculation is
  1. Pick two points on the line and determine their coordinates.
  2. Determine the difference in y-coordinates for these two points (rise).
  3. Determine the difference in x-coordinates for these two points (run).
  4. Divide the difference in y-coordinates by the difference in x-coordinates (rise/run or slope).
Now look back at the previous example:

The equation of a line with slope -2 and the point (4,6)
y-6=-2(x-4)
y-6=-2x+8
2x+y=14
 
Now let us find the slope of (-2,-3) and (3,-1) the graph of which is shown below:
pointslope.gif
First, find the slope using the points (- 2, 3) and (3, - 1):



m=
3-(-1)
-2-3

=
4
-5


=

-
4
5		 = -0.80


Next, pick a point - for example, (- 2, 3). Using this point, x = - 2 and y = 3. Therefore, the equation of this line is:


y - 3 = -		 4
5

 (x - (- 2)), which is equivalent to:
 


y - 3 = -		 4
5

 (x + 2).   Check the graph using the point.

So,  (3, -1): -1 - 3 = - (3 + 2), which is the same as the graph shown above.

 
Now let's look at the same application of the point slope to the vectors that I have been teaching you about.  Remember that a vector is both magnitude (distance) and direction.  Calculate (3,4) and (-2,-3).  The vector will be the direction (m = slope) and the length of the line between (3,4) and (-2,-3) will be it's magnitude.

To calculate the length of this magnitude:

(3,4) and (-2,-3),

We can use the Pythagorean theorem:

a2 +b2; = √c2; We can rewrite this using the absolute values of y + y', and x +x':

a = y + y' = |-1+(-3)|2,

b = x + x' = |-2+3|2,

c = 16 + 25 = √41

Remember to change all of the values in this step to absolute.  There would be no use calculating the lengths of sides a (y axis) and b (x axis) if these were not treated as absolute values.
 
So that the magnitude (distance or length of the hypoteneus) is equal to 6.40 and the slope (direction) is equal to -0.80.

For a method of vector addition that does not rely on the Pythagorean theorem see this page.


File translated from TEX by TTH, version 3.70.
On 20 Dec 2005, 06:29.