2. The Straight Line
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Slope-Intercept Form of a Straight Line
The slope-intercept form (otherwise known as "gradient, y-intercept" form) of a line is given by:
y = mx + b
This tells us the slope of the line is m and the y-intercept of the line is b.
Example:
The line y = 2x + 4 has
- slope m = 2 and
- y-intercept b = 4.
We do not need to set up a table of values to sketch this line. Starting at the y-intercept (y = 4), we sketch our line by going up 2 units for each unit we go to the right (since the slope is 2 in this example). To find the x-intercept, we let y = 0.
2x + 4 = 0
x = -2
We notice that this is a function. That is, each value of x that we have gives one corresponding value of y.
See more on Functions and Graphs.
Point-Slope Form of a Straight Line
We need other forms of the straight line as well. A useful form is the point-slope form (or point - gradient form). We use this form when we need to find the equation of a line passing through a point (x1, y1) with slope m:
y − y1 = m(x − x1)
Example:
Need Graph Paper?
Find the equation of the line that passes through (-2, 1) with slope of -3.
This LiveMath document lets us play with this idea. Let's do this first.
Now for the normal answer:
General Form
Another form of the straight line which we come across is general form:
Ax + By + C = 0
It can be useful for drawing lines by finding the y-intercept (put x = 0) and the x-intercept (put y = 0).
We also use General Form when finding Perpendicular Distance from a Point to a Line.
Example:
Draw the line 2x + 3y + 12 = 0.
Exercises
1. What is the equation of the line perpendicular to the line joining (4, 2) and (3, -5) and passing through (4, 2)?
[Need a reminder? See the section on Slopes of Perpendicular Lines.]
2. If 4x − ky = 6 and 6x + 3y + 2 = 0 are perpendicular, what is the value of k?
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