PARAMETRIC CURVES
ASSIGNMENT 10
BY: SHARREN M. THOMAS
The problem:
Write parametric equations
of a line segment through (7, 5) with slope of 3. Graph the line segment
using your equations.
Recall the point-slope
form, an equation for the line is
y - y1 = m ( x -
x1).
If we let x = t, the y - y1
= m (t - x1) and we obtain the parametrization
x = t, y - y1 = m (t - x1 ) ;
t in R.
We can obtain another
parametrization for the line if we let x - x1 = t in R.
In this case y - y1
= mt, and we have
x = x1 + t, y = y1 + mt; t in R
In order to begin this
problem. In general, a pair of parametric equations is a pair of
continuous functions that define the x- and y- coordinates of a point in a
coordinate plane in terms of a third variable, such as t, called the parameter.
Thus, a parametric curve in
the plane is a pair of functions
x = f (t)
y = g (t)
where the two continuous
functions define ordered pairs (x, y). Substituting into : x = x1
+ t, y = y1 + mt; t in R we get:
When I put these back in
point-slope form I obtain the following:
y - 5 = 3 (x - 7)
y = 3x - 21 + 5
y = 3x - 16
So, both the parametric
equations and y = 3x - 16 should graph the same line. See below.
As you can see above these
are the same lines.