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 - y_{1} = m ( x -
x_{1}).

If we let x = t, the y - y_{1
}= m (t - x_{1}) and we obtain the parametrization

x = t, y - y_{1} = m (t - x_{1 })_{ };
t in R.

We can obtain another
parametrization for the line if we let x - x_{1} = t in R.

In this case y - y_{1}
= mt, and we have

x = x_{1} + t, y = y_{1} + 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 = x_{1}
+ t, y = y_{1} + 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.