https://r2.erweima.ai/mathbot/image/17260265008722n7av4ei.jpg

The problem involves converting parametric equations into their explicit or implicit forms.

Let's work through each case step by step:

### I. Parametric Equations:
\[
f = \begin{cases} 
x = t^2 - 2t \\
y = t + 1 
\end{cases}
\]
We need to eliminate \(t\) to find the equation involving only \(x\) and \(y\).

1. Solve for \(t\) from the second equation:
\[
y = t + 1 \implies t = y - 1
\]

2. Substitute \(t = y - 1\) into the equation for \(x\):
\[
x = (y - 1)^2 - 2(y - 1)
\]
Expand and simplify:
\[
x = (y^2 - 2y + 1) - 2(y - 1) = y^2 - 2y + 1 - 2y + 2
\]
\[
x = y^2 - 4y + 3
\]
Thus, the equation is:
\[
x = y^2 - 4y + 3
\]

There is no option exactly matching this, but you can check for typographical differences or match it to one of the closest options.

### II. Parametric Equations:
\[
f = \begin{cases} 
x = 4 \cos \theta \\
y = 4 \sin \theta 
\end{cases}
\quad 0 \leq \theta \leq 2\pi
\]

These equations represent a parametric form of a circle. To derive the implicit form, recall that \(\cos^2 \theta + \sin^2 \theta = 1\).

1. Solve for \(\cos \theta\) and \(\sin \theta\) from each parametric equation:
\[
\cos \theta = \frac{x}{4}, \quad \sin \theta = \frac{y}{4}
\]

2. Use the Pythagorean identity:
\[
\left( \frac{x}{4} \right)^2 + \left( \frac{y}{4} \right)^2 = 1
\]
\[
\frac{x^2}{16} + \frac{y^2}{16} = 1
\]

Thus, the equation is:
\[
\frac{x^2}{16} + \frac{y^2}{16} = 1
\]
This matches option (a).

### III. Parametric Equations:
\[
f = \begin{cases} 
x = 3 - 4t \\
y = 2 - 3t 
\end{cases}
\]

We need to eliminate \(t\) to find the equation involving only \(x\) and \(y\).

1. Solve for \(t\) from the first equation:
\[
x = 3 - 4t \implies t = \frac{3 - x}{4}
\]

2. Substitute this into the equation for \(y\):
\[
y = 2 - 3\left(\frac{3 - x}{4}\right)
\]
Simplify:
\[
y = 2 - \frac{9 - 3x}{4} = 2 - \frac{9}{4} + \frac{3x}{4}
\]
\[
y = \frac{8}{4} - \frac{9}{4} + \frac{3x}{4} = \frac{3x - 1}{4}
\]
Thus, the equation is:
\[
4y = 3x - 1
\]
This matches option (a).

---

### Summary of Answers:
1. **I**: None of the provided options exactly match.
2. **II**: Option (a) \(\frac{x^2}{16} + \frac{y^2}{16} = 1\).
3. **III**: Option (a) \(4y = 3x - 1\).

Would you like more detailed explanations for any of these steps?

### Related Questions:
1. How can we generalize the process of eliminating a parameter \(t\) in parametric equations?
2. Can we always derive a circle equation from trigonometric parametric equations like in problem II?
3. What happens if the parametric equations represent a different conic section, like an ellipse or parabola?
4. How can we verify the domain restrictions of \(\theta\) or \(t\) when converting parametric equations?
5. What are other methods of deriving implicit equations from parametric forms?

### Tip:
Always remember that when solving parametric equations, isolating the parameter (like \(t\) or \(\theta\)) from one equation and substituting it into the other is a reliable strategy for elimination.

Based on the parametric equations provided in the image, determine the equivalent explicit or implicit equation.

Solve parametric equations like x = t^2 - 2t and y = t + 1 by eliminating the parameter t to find the corresponding explicit or implicit equations. This problem covers algebra and trigonometry, making it suitable for students in Grades 10-12.

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