37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.

The line that passes through the points P(1, 1) and Q(3, 5), oriented in the direction of increasing x

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Identify the two given points: P(1, 1) and Q(3, 5). These points lie on the line we want to parametrize.

Calculate the direction vector \( \vec{d} \) of the line by subtracting the coordinates of P from Q: \( \vec{d} = (3 - 1, 5 - 1) = (2, 4) \). This vector points in the direction of increasing x.

Set up the parametric equations using point P as the initial point and \( t \) as the parameter: \( x(t) = 1 + 2t \) and \( y(t) = 1 + 4t \). Here, \( t \) represents how far along the line you move from point P.

Determine the interval for \( t \) based on the problem context. Since the line extends infinitely in both directions, \( t \) can be any real number, but to keep the orientation in the direction of increasing x, \( t \) should be chosen such that \( x(t) \) increases as \( t \) increases, so \( t \in \mathbb{R} \).

Summarize the parametric form: \( x(t) = 1 + 2t \), \( y(t) = 1 + 4t \), with \( t \in (-\infty, \infty) \). This fully describes the line passing through P and Q oriented in the direction of increasing x.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Parametric Equations of a Line

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. For a line through two points, the equations are derived by setting x and y as linear functions of t, representing movement from one point to another.

Vector Form and Direction of a Line

The direction vector of a line is found by subtracting the coordinates of the initial point from the terminal point. This vector determines the line's orientation, and the parameter t scales this vector to generate all points along the line.

Parameter Interval and Orientation

The parameter interval defines the range of t values for which the parametric equations are valid. Choosing an interval consistent with the direction of increasing x ensures the line is oriented correctly, reflecting the problem's requirement.

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.The line that passes through the points P(1, 1) and Q(3, 5), oriented in the direction of increasing x