Let's go through the steps needed to solve the problem shown in the image:

Part (a): Eliminate the parameter to find a Cartesian equation of the curve.

You are given:

• $x = |t|$
• $y = 8 - |t|$

To eliminate the parameter $t$, observe that both $x$ and $y$ involve the absolute value of $t$. From the first equation, we have $|t| = x$. Substituting this into the second equation gives: $y = 8 - |t| = 8 - x$ Thus, the Cartesian equation of the curve is: $y = 8 - x$ This is a straight line equation.

Part (b): Sketch the curve and indicate the direction of the curve.

Since $t$ is involved with absolute values, the parameter $t$ will cover both positive and negative values. Let's analyze how the curve is traced as $t$ increases:

• When $t = 0$, $x = 0$ and $y = 8$, so the curve starts at the point $(0, 8)$.
• As $t$ increases (positive values), $x = t$ and $y = 8 - t$, meaning the curve moves downward and right.
• As $t$ becomes negative, $x = -t$, meaning $x$ becomes positive but increasing. The curve moves downward and right again, meeting at the same line $y = 8 - x$.

In terms of the direction, we can see that as $t$ increases from negative values to positive, the curve traces from left to right.

Choice for sketch:

• The correct choice is Choice D, which shows the curve traced starting from $(0, 8)$ and symmetrically moving down and outward in both directions.

Let me know if you'd like more detailed steps, or if you have further questions! Here are some additional questions to deepen your understanding:

1. How would the equation change if the parameter involved a quadratic term in $t$ instead of absolute values?
2. What would the graph look like if $y = 8 + |t|$ instead of $y = 8 - |t|$?
3. How does the presence of $|t|$ affect the shape of the curve?
4. What would be the domain and range of the function $y = 8 - x$?
5. How would the graph behave if you introduced a vertical or horizontal shift?

Tip:

When dealing with parametric equations, always analyze how the parameter behaves over its range to understand the graph's direction and shape.