Mathematics

Problem

## 66. Group Activity Parametrization of Lines Consider the parametrization

x=a t+b, \quad y=c t+d,

where

a

and

c

are not both zero. (a) Graph the curve for

a=2, b=3, c=-1

, and

d=2

. (b) Graph the curve for

a=3, b=4, c=1

, and

d=3

. (c) Writing to Learn Eliminate the parameter

t

and write an equation in

x

and

y

for the curve. Explain why its graph is a line. (d) Writing to Learn Find the slope,

y

-intercept, and

x

-intercept of the line if they exist. If not, explain why not. (e) Under what conditions will the line be horizontal? Vertical?

Intuitive explanation

A parametric equation traces out a curve by letting a parameter $t$ vary—think of it as a moving point whose position depends on time. Here, both $x$ and $y$ are linear functions of $t$ , so as $t$ changes, the point moves along a straight path. To understand the line's shape and properties, you'll eliminate $t$ by solving one equation for $t$ and substituting into the other, revealing the familiar $y = mx + b$ form (or a vertical line). The coefficients $a$ and $c$ control the line's direction and steepness, while $b$ and $d$ set where the path starts.

Intuitive explanation

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