Finding Parametric Equations and Tangent Lines

Find parametric equations for the given curve.

Line through (1,-2) with slope 3

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Recall that a parametric equation for a line can be expressed as \(x = x_0 + at\) and \(y = y_0 + bt\), where \((x_0, y_0)\) is a point on the line and \((a, b)\) is a direction vector parallel to the line.

Identify the given point on the line as \((1, -2)\), so \(x_0 = 1\) and \(y_0 = -2\).

Use the slope of the line, which is 3, to find the direction vector. Since slope \(m = \frac{b}{a} = 3\), you can choose \(a = 1\) and \(b = 3\) for simplicity.

Write the parametric equations using the point and direction vector: \(x = 1 + 1 \cdot t\) and \(y = -2 + 3 \cdot t\).

These parametric equations describe the line passing through \((1, -2)\) with slope 3, where \(t\) is the parameter.

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Parametric Equations

Parametric equations express a curve by defining both x and y coordinates as functions of a third variable, usually t. This allows representation of curves that are difficult to describe with a single function y = f(x). For example, a line can be represented as x = x0 + at and y = y0 + bt, where (x0, y0) is a point on the line and (a, b) relates to the direction.

Equation of a Line Using Point-Slope Form

The point-slope form of a line is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. This form is useful for quickly writing the equation of a line when a point and slope are known. It can be rearranged or converted into parametric form for further analysis.

Tangent Lines and Their Slopes

A tangent line to a curve at a point touches the curve without crossing it and has the same slope as the curve at that point. For parametric curves, the slope of the tangent line is found by dy/dx = (dy/dt) / (dx/dt). Understanding tangent lines helps in analyzing the behavior and direction of curves.

Finding Parametric Equations and Tangent LinesFind parametric equations for the given curve.Line through (1,-2) with slope 3