The problem involves graphing the parametric equations $x = \frac{1}{4}t$ and $y = 3t - 8$, and then converting these parametric equations to an equivalent rectangular equation.

Step 1: Eliminate the parameter $t$

We need to express $x$ and $y$ in terms of one another by solving for $t$ from one equation and substituting it into the other.

1. From $x = \frac{1}{4}t$, solve for $t$: $t = 4x$

2. Substitute $t = 4x$ into $y = 3t - 8$: $y = 3(4x) - 8$ Simplify: $y = 12x - 8$

Step 2: Determine the range of $x$ and $y$

The parameter $t$ is given to be in the range $-3 \leq t \leq 3$.

• Using $t = 4x$: $-3 \leq 4x \leq 3 \implies -\frac{3}{4} \leq x \leq \frac{3}{4}$

• Calculate the corresponding $y$-values using $y = 12x - 8$: $\text{When } x = -\frac{3}{4}, \, y = 12\left(-\frac{3}{4}\right) - 8 = -9 - 8 = -17$ $\text{When } x = \frac{3}{4}, \, y = 12\left(\frac{3}{4}\right) - 8 = 9 - 8 = 1$

So, the line $y = 12x - 8$ is graphed for $-\frac{3}{4} \leq x \leq \frac{3}{4}$, and $y$ ranges from $-17$ to $1$.

Step 3: Match with the given graphs

Looking at the graphs:

• The correct graph should represent a line with a slope of 12, passing through $(x, y) = (0, -8)$ (the $y$-intercept), and restricted to the intervals derived above.

Correct Answer:

The correct graph matches Option C. It shows the line segment $y = 12x - 8$ with the correct slope and bounds.

Would you like a more detailed explanation of any step? Here are related questions to expand on this concept:

1. How do you graph a line given a parametric equation?
2. What are other methods to eliminate the parameter $t$ in parametric equations?
3. Can all parametric equations be converted to rectangular form?
4. How do parametric restrictions influence the graph of a line?
5. What is the significance of slope and $y$-intercept in linear equations?

Tip: Always check the bounds of your parametric equations to correctly limit the graph to the defined segment.