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:
|y - k = m(x - h)|
where m is the slope of the line and (h, k) is a point on the line (any point works).
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 (h, k). It does not matter which point you pick, as long as it is 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, h, and k. Check your equation by picking a point on the line--not the point you chose as (h, k)--and confirming that it satisfies the equation.
Example 1: Write an equation of the following line in point-slope form:
First, find the slope using the points (- 2, 3) and (3, - 1): m = = = - .
Next, pick a point -- for example, (- 2, 3). Using this point, h = - 2 and k = 3.
Therefore, the equation of this line is y - 3 = - (x - (- 2)), which is equivalent to y - 3 = - (x + 2).
Check using the point (3, -1): -1 - 3 = - (3 + 2) ? Yes.
Example 2: Write an equation of the line which passes through (3, 4) and has slope m = 5.
h = 3 and k = 4. y - 4 = 5(x - 3)
Example 3: Write an equation of the line which is parallel to the line y = 3x + 2 and passes through (- 1, 2).
m = 3, h = - 1, and k = 2.
The equation of the line is y - 2 = 3(x + 1).
Example 4: Write an equation of the line which is perpendicular to the line y - 8 = 2(x + 2) and passes through (7, 0).
The slope is the opposite reciprocal of 2: m = - . h = 7 and k = 0.
The equation of the line is y - 0 = - (x - 7), which is equivalent to y = - (x - 7).
Example 5: Write an equation of the line with slope m = 4 that passes through the point (0, 3).
m = 4, h = 0, and k = 3.
The equation of the line is y - 3 = 4x. If we move -3 to the other side--y = 4x + 3--we get the equation in slope-intercept form.