Given two
different points we can always draw a line segment between them. We can also sketch a line given just one point and a direction. How? The
instructions are ‘hiding in plain view’ in the equation y = mx + c. Take, for
example y = ½ x – 3.
Please do not make a table of values for x and
y.
Tables are
slow and inefficient. Remember, we only
need one point! Which point? The
yintercept. For the equation y = ½ x –
3, start at the point (0, c) = (0, 3).
Now what? From the gradient, m =
½, we find that the rise is 1 and the
run is 2.
If the
gradient is written as a fraction (½, 23/7, 4/3, …) the numerator is the
‘rise’ and the denominator is the run.
If, however, the gradient is written as a decimal number (0.5, 1.6, 
3.9, …) the given number is the ‘rise’ and the run is always ‘1’. Positive runs
go to the right; negative runs to the left.
Positive rises go up; negative rises go down.
With
preliminaries out of the way, here is the line: Start at the yintercept,
sketch the run (running parallel to the xaxis) and then sketch the rise
(rising parallel to the yaxis). Label
the end point, P. Take a ruler and
connect the yintercept and point P.
You can
explore with the Geogebra file “mx_plus_c” to vary intercept c and gradient m
using the respectively labeled sliders.
Turn on the trace function for
point P by right clicking on the point and selecting “Trace On”. Move the c and m sliders to choose your
desired y intercept and gradient. After you have chosen these parameters, go
the pulldown menu, View and click “Refresh Views”. Finally, go to the x_{P} slider and drag that button.
What happens?
The rise
and run are continually refreshed, so the screen only shows the latest values,
but we see all the previous values of point P.
The locus of point P is the required line. Ta Dah!
Conclusion:
For many of
you this essay merely restated the obvious.
However, the next installment will use very similar ideas to explain the
vector equation of a line:
p = q + kv .
Appendix:
Geogebra Construction
No.

Name

current value


1

Number, m

a
slider variable

m = 1.7

2

Number, c

a
slider variable

c = 1

3

Point, O

the
origin

O = (0, 0)

4

Point, I

(0, c)

I = (0,
1)

5

Vector,
vector_{c}

Vector[point,
point] = Vector[O, I]

vector_{c} = (0, 1)

6

Number, x_{p}

a
slider variable

x_{p} = 2.8

7

Vector run

Vector[point,
point] = Vector[(0, c), (xp, c)]

run =
(2.8, 0)

8

Vector
rise

Vector[point,
point] =Vector[(xp, c), (xp, c + xp times m)]

rise = (0,
4.76)

9

Point P

vectorc + run + rise

P = (2.8,
3.76)

A copy of the applet is stored on http://www.geogebratube.org/
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