Problem 29 - Precalculus Mathematics for Calculus

Question

Find parametric equations for the line with the given properties. Slope \(\frac{1}{2},\) passing through \((4,-1)\)

Step-by-Step Solution

Answer

The parametric equations are \( x = 4 + 2t \) and \( y = -1 + t \).

Step 1: Understand the properties of the line

The line has a slope \( m = \frac{1}{2} \) and passes through the point \( (4, -1) \). We need to express the equation of this line in parametric form.

Step 2: General form of parametric equations for a line

The parametric equations of a line can be written as: \( x = x_0 + at \) and \( y = y_0 + bt \), where \( (x_0, y_0) \) is a point on the line, \( a \) and \( b \) are coefficients related to the slope, and \( t \) is a parameter.

Step 3: Determine coefficients using point-slope relationship

Given that the slope \( m = \frac{1}{2} \), we know that \( b = \frac{a}{2} \). Choose \( a = 2 \) (for simplicity) and consequently \( b = 1 \). This satisfies the slope condition.

Step 4: Write the parametric equations

Using the point \( (4, -1) \) and the chosen coefficients \( a = 2 \) and \( b = 1 \), we can write the parametric equations: \( x = 4 + 2t \) and \( y = -1 + t \).

Key Concepts

Line Equation

The line equation is a mathematical representation that precisely defines a straight line on a plane. There are various forms of line equations, but fundamentally, they express the relationship between the x-coordinate and y-coordinate of points along the line. One common form is the slope-intercept form, represented as \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. This form highlights how the line tilts or rises across the plane by indicating how much y moves with respect to x. Line equations are foundational in geometry and algebra as they help locate any point along the line and understand its trajectory.

Slope

The slope of a line measures its steepness and direction. It is expressed as the ratio of the rise over the run between two points on the line—literally how much you "up" or "down" you go for a given "across." Mathematically, given two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope \( m \) is calculated using the formula

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

The slope helps in determining how the line behaves:

• A positive slope means the line goes upwards as it moves from left to right.
• A negative slope implies it goes downwards.
• A zero slope indicates a flat line, while an undefined slope is a vertical line.

Parametric Form

Parametric equations provide a way to represent a line using parameters, allowing for comprehensive flexibility and description of a curve's position on a coordinate plane. For lines, they are in the form:

• \( x = x_0 + at \)
• \( y = y_0 + bt \)

Here, \((x_0, y_0)\) is a specific point through which the line passes.

• \( a \) and \( b \) are constants related to the slope of the line, where \( b = \frac{a}{m} \).
• \( t \) is the parameter, often representing time, showing how the line progresses over its length.

This form allows us to easily trace every point on a line by varying \( t \), beneficial in both mathematical modeling and computer graphics.

Point-Slope Form

Point-slope form is a tool for writing the equation of a line when you know one point it passes through and the line's slope. The equation is:

• \( y - y_1 = m(x - x_1) \)

Here, \( m \) is the slope, and \((x_1, y_1)\) is a point on the line. This form is especially useful because you only need these two pieces of information—one point and the slope—to express the equation of a line.

• It provides a direct connection between coordinates and their use in calculations.
• It swiftly translates into other line forms, like slope-intercept by solving for \( y \).

This straightforward approach allows you to quickly derive the line's equation, making it a favorite in algebra for tackling line-related problems.