2. The Straight Line

Slope-Intercept Form of a Straight Line

graph of y= mx + b

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 unit we go to the right (since the slope is `2` in this example).

graph of 2x - 4

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.

Continues below

Point-Slope Form of a Straight Line

graph showing point-slope form

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(xx1)

Example 2

Find the equation of the line that passes through `(-2, 1)` with slope of `-3`.

General Form of a Straight Line

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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.

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`?

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):

double cone

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.

conic section - straight line