2. The Straight Line
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 1
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 `1` 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 2
Find the equation
of the line that passes through `(-2, 1)` with slope of `-3`.
Answer
We use:
`y-y_1=m(x-x_1)`
Here,
`x_1= -2``y_1= 1`
`m = -3`
So the required equation is:
`y-1=-3(x-(-2)=-3x-6`
`y=-3x-5`
We have left it in slope-intercept form. We can see the slope is -3 and the y-intercept is -5.
General Form of a Straight Line
Need Graph Paper?
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 3
Draw the line 2x + 3y + 12 = 0.
Answer
If `x = 0`, we have: `3y + 12 = 0`, so `y = -4`.
If `y = 0`, we have: `2x + 12 = 0`, so `x = -6`.
So the line is:
Note that the y-intercept is `-4` and the x-intercept is `-6`.
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.]
Answer
The line joining `(4, 2)` and `(3, -5)` has slope `m=(-7)/(-1)=7` and is shown as a green dotted line.
We need to find the equation of the magenta (pink) line.
The line perpendicular to the green dotted line has slope `-1/7.`
The line through `(4, 2)` with slope `-1/7` has equation:
`y-2=-1/7(x-4)`
`=-x/7+4/7`
`y=-x/7+2 4/7`
2. If `4x − ky = 6` and `6x + 3y + 2 = 0` are perpendicular, what is the value of `k`?
Answer
(2) The slope of 4x − ky = 6 can be calculated by re-expressing it in slope-intercept form:
`y=4/kx-6/k`
So we see the slope is `4/k`.
The slope of `6x + 3y + 2 = 0` can also be calculated by re-expressing it in slope-intercept form:
`y=(-6)/3x-2/3=-2x-2/3`
So we see the slope is `-2`.
For the lines to be perpendicular, we need
`4/kxx-2=-1`
This gives `k = 8`.
The resulting line is `4x-8y=6`, which we can simplify to `2x-4y=3.` Here's the graph of the situation:
Perpendicular lines
Conic section: Straight line
Each of the lines and curves in this chapter are conic sections, which means the curves are formed when we slice a cone at a certain angle.
How can we obtain a straight line from slicing a cone?
We start with a double cone (2 right circular cones placed apex to apex):
If we slice the double cone by a plane just touching one edge of the double cone, the intersection is a straight line, as shown.