91Ó°ÊÓ

The x-intercept is a crucial point where a curve crosses the x-axis. To find this in parametric equations, we look for the value of the parameter that makes the y-coordinate zero. In our problem, the parametric equations are given as:

• \( x = t + 2 \)
• \( y = 3 + \frac{1}{2}t \)

To determine the x-intercept, we set \( y = 0 \). Solving the equation \( 3 + \frac{1}{2}t = 0 \), we approach it step-by-step:

• Move the 3 over to the right side: \( \frac{1}{2}t = -3 \).
• Multiply both sides by 2 to solve for \( t \): \( t = -6 \).

Now we need to find the x-value using \( x = t + 2 \):

• Substitute \( t = -6 \) resulting in \( x = -6 + 2 = -4 \).

Therefore, the x-intercept of the curve is at \((-4, 0)\). This point signifies when the curve meets the x-axis.

The y-intercept of a curve is the point where it crosses the y-axis. To find this using parametric equations, we need to identify the parameter value that results in an x-coordinate of zero. For our problem, the parametric equations are:

• \( x = t + 2 \)
• \( y = 3 + \frac{1}{2}t \)

To find the y-intercept, we set \( x = 0 \) and solve \( t + 2 = 0 \):

• Solve for \( t \): \( t = -2 \).

We then substitute this back into the y-equation:

• Calculate \( y \) using \( y = 3 + \frac{1}{2}(-2) \).
• This simplifies to \( y = 3 - 1 = 2 \).

Thus, the y-intercept is at \((0, 2)\), marking the point where the curve crosses the y-axis. This point is important in understanding the initial behavior of the curve on a graph.

When sketching curves described by parametric equations, like the ones in this problem, it's essential to convert a set of parametric points into a visual representation. First, let’s identify key points by substituting a range of \( t \) values into the parametric equations:

• \( x = t + 2 \)
• \( y = 3 + \frac{1}{2}t \)

For \( t = -3, -2, -1, 0, 1, 2, 3 \), calculating gives:

• \((-3, 1.5)\)
• \((-2, 2)\)
• \((-1, 2.5)\)
• \((0, 3)\)
• \((1, 3.5)\)
• \((2, 4)\)
• \((3, 4.5)\)

Plot these points on a coordinate plane. As you do, you'll notice they form a straight line with positive slope. The curve's direction indicates its orientation as \( t \) increases, which in this case is moving rightwards.Understanding the flow is key to accurately sketching parametric curves. The entire process helps us to visualize how the equations describe the movement of the curve over time. This is a fundamental skill that assists in interpreting and predicting the behavior of more complex systems.