https://file.aiquickdraw.com/mathbot/file/1733548439518hku1fmuz.jpg

From the image, we have the parametric equations:

\[
x = 4t, \quad y = t + 6
\]

### **Part (a): Sketching the curve**
To determine the direction of the curve as \( t \) increases:
1. Solve for a relationship between \( x \) and \( y \) by eliminating \( t \).
   - From \( x = 4t \), solve for \( t \): \( t = \frac{x}{4} \).
   - Substitute \( t = \frac{x}{4} \) into \( y = t + 6 \):
     \[
     y = \frac{x}{4} + 6
     \]
     This is the equation of a straight line with slope \( \frac{1}{4} \) and \( y \)-intercept \( 6 \).
     
2. The curve's direction is determined by increasing \( t \). As \( t \) increases:
   - \( x = 4t \): \( x \) increases.
   - \( y = t + 6 \): \( y \) increases as well.

Thus, the direction of the curve is from bottom-left to top-right, matching the **lower-right graph in the image**.

---

### **Part (b): Find the rectangular-coordinate equation**
From above, the rectangular equation is:

\[
y = \frac{x}{4} + 6
\]

This is the required rectangular-coordinate equation.

---

Let me know if you'd like a detailed breakdown of the solution! 

### Questions for further exploration:
1. What are the steps for solving a parametric equation when the parameter appears in nonlinear terms?
2. How can you confirm the direction of a parametric curve numerically?
3. What is the geometric interpretation of eliminating a parameter from equations?
4. How does the slope in the rectangular equation relate to the parametric equations?
5. Can we parameterize this line using different equations?

### Tip:
When sketching parametric curves, always calculate the direction by considering changes in \( x \) and \( y \) as \( t \) increases!

The user is asked to solve a parametric equation, specifically sketching the curve and finding the corresponding rectangular-coordinate equation.

Learn how to solve parametric equations and find the corresponding rectangular-coordinate equation. This detailed solution covers concepts such as parametric equations, graphing, and eliminating parameters, suitable for students in Grades 10-12.

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