To find a set of parametric equations for [tex]\(y = \frac{(x-3)^2}{2}\)[/tex], we can introduce a parameter, say [tex]\(t\)[/tex], and express both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in terms of this parameter.
Let's perform this step-by-step:
1. Parameter Representation:
Choose a parameter [tex]\(t\)[/tex]. To make the equations straightforward, let's set [tex]\(x = t\)[/tex]. Therefore, parameter [tex]\(t\)[/tex] represents the [tex]\(x\)[/tex]-coordinate.
2. Expressing [tex]\(y\)[/tex] in terms of [tex]\(t\)[/tex]:
Substitute [tex]\(x = t\)[/tex] into the given equation [tex]\(y = \frac{(x-3)^2}{2}\)[/tex].
[tex]\[
y = \frac{(t - 3)^2}{2}
\][/tex]
3. Forming the Parametric Equations:
Now, we have both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] expressed in terms of the parameter [tex]\(t\)[/tex]:
[tex]\[
x = t
\][/tex]
[tex]\[
y = \frac{(t - 3)^2}{2}
\][/tex]
Therefore, the set of parametric equations for the given curve [tex]\(y = \frac{(x-3)^2}{2}\)[/tex] is:
[tex]\[
\begin{cases}
x = t \\
y = \frac{(t - 3)^2}{2}
\end{cases}
\][/tex]
Here, [tex]\(t\)[/tex] can take any real number, representing different points on the curve.
