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Curve sketching with vector-valued functions is a task where we visually represent the path traced by the function on a coordinate plane. In the case of the function \( \mathbf{F}(t) = t \mathbf{i} \), which only includes the \( i \) component, the sketching process becomes straightforward. The key is to understand that since the function moves entirely along the x-axis due to the dependence solely on \( t \), our task reduces to drawing a line segment along the x-axis.

The line segment stretches between two points determined by the parameter values of \( t \). Here, as you sketch, you’ll move from left to right, starting at the point where \( t = -1 \), resulting in the point \((-1, 0)\). You’ll proceed to draw a straight line to the endpoint \( \left(\frac{1}{2}, 0\right) \) where \( t = \frac{1}{2} \). Remember to consider the nature of the path the function traces, which is linear in this scenario, starting from the left at \((-1, 0)\) and ending at the right at \(\left(\frac{1}{2}, 0\right)\).

• Identify starting and ending points for your sketch based on \( t \).
• Since there's no \( j \) component, focus on the x-axis for movement.
• Linear path means a straight line on the x-axis.

The parameter range is crucial as it informs us about the extent of the curve's trace. In vector-valued functions, this range denotes the values \( t \) can take, directly affecting the potential positions of the vector on the graph.

For our particular function \( \mathbf{F}(t) = t \mathbf{i} \), the parameter range is given as \(-1 \leq t \leq \frac{1}{2}\). This range indicates that, for our sketch, the curve will be traced from \( t = -1 \) to \( t = \frac{1}{2}\). Consequently, the vector starts its journey on the x-axis at position \( x = -1 \) (when \( t = -1 \)) and ends at \( x = \frac{1}{2} \) (when \( t = \frac{1}{2} \)).

Understanding the parameter range helps us capture the full span of the curve on the graph:

• Determine the start and end points of your graph.
• The range of \( t \) values guides the initial and final x-coordinates.
• No dependence means uniform progression along the defined stretch on the x-axis.

Determining the direction of the curve is a vital element in fully communicating how a vector-valued function behaves. In this particular function, the direction is along the x-axis due to the absence of a \( j \) component. Hence, there is no vertical change as \( t \) varies. We entirely focus on the shift in horizontal direction.

The parameter \( t \) increases from \(-1\) to \(\frac{1}{2}\), leading to a rightward movement on our sketch. One identifies this by observing the value change in \( \mathbf{F}(t) = t \mathbf{i} \). As \( t \) progresses from \(-1\) to \( \frac{1}{2} \), the direction of the curve follows suit, essentially moving from left to right along the x-axis. This observation is visually represented on the sketch by drawing an arrow. Such arrows depict the flow and provide clarity to otherwise static graph representations.

• Direction is determined by the sign and range of \( t \).
• The function progresses in a linear, horizontal manner.
• Indicate movement with arrows to show increasing \( t \) direction.