To sketch the curve defined by the parametric equations $x = 3t - 4$ and $y = 6t + 2$, we need to eliminate the parameter $t$ and express $y$ as a function of $x$.

Step 1: Solve for $t$ in terms of $x$

From the equation: $x = 3t - 4$ we can solve for $t$: $t = \frac{x + 4}{3}$

Step 2: Substitute $t$ in the equation for $y$

Now, substitute $t = \frac{x + 4}{3}$ into the equation $y = 6t + 2$: $y = 6\left(\frac{x + 4}{3}\right) + 2$ Simplify this: $y = 2(x + 4) + 2$ $y = 2x + 8 + 2$ $y = 2x + 10$

So, the Cartesian equation of the curve is: $y = 2x + 10$

This equation represents a straight line with a slope of $2$ and a y-intercept of $10$.

Step 3: Determine the direction of increasing $t$

To find the direction of increasing $t$, we can analyze the behavior of $x$ and $y$ as $t$ increases:

• As $t$ increases, $x = 3t - 4$ also increases.
• Similarly, $y = 6t + 2$ increases as $t$ increases.

Thus, the direction of increasing $t$ is from left to right along the line $y = 2x + 10$.

Summary

The sketch of the curve will be a straight line given by $y = 2x + 10$, with the direction of increasing $t$ moving from left to right along the line.

Would you like to see more details on graphing this line, or do you have any questions?

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1. How would we approach this if the equations were nonlinear?
2. How do we handle cases where eliminating the parameter isn't straightforward?
3. What if the direction of $t$ were reversed?
4. How can we check our result for parametric equations using a graphing calculator?
5. What are alternative methods for sketching parametric curves?

Tip: When converting parametric equations, always check if simplifying assumptions (like linearity) apply to make the process quicker.