Problem 49 - A Graphical Approach to College Algebra

Question

Give two parametric representations of the line through the point \(\left(x_{1}, y_{1}\right)\) with slope \(m\)

Step-by-Step Solution

Answer

\((x, y) = (x_1 + t, y_1 + mt)\) and \((x, y) = (x_1 - s, y_1 - ms)\).

Step 1: Recall the point-slope form of a line

The point-slope form of a line is given by the equation \( y - y_1 = m(x - x_1) \). This form indicates a line with slope \( m \) passing through the point \( (x_1, y_1) \).

Step 2: Define parametric equations for the line

Parametric equations express each of the coordinates \( x \) and \( y \) separately in terms of a parameter, typically \( t \). For a line with slope \( m \) through the point \( (x_1, y_1) \), one parametric representation is: \[ x = x_1 + t \] \[ y = y_1 + mt \] where \( t \) is a parameter that varies over all real numbers.

Step 3: Alternative parametric representation

We can also write a different parametric representation by considering another parameter, say \( s = -t \): \[ x = x_1 - s \] \[ y = y_1 - ms \] where \( s \) also varies over all real numbers.

Key Concepts

Point-Slope Form

The point-slope form is a simple yet powerful way to describe a line with a given slope passing through a known point. It is expressed as \( y - y_1 = m(x - x_1) \), where

• \( m \) represents the slope of the line.
• \((x_1, y_1)\) is the known point through which the line passes.

This form is particularly useful when you have a specific point and a slope, but not the y-intercept readily available.
Unlike the standard form \( y = mx + b \), which requires pinpointing the y-intercept \( b \), the point-slope form is more flexible in settings where the point isn't obvious from a graph.
It provides a straightforward approach to identify points along a line without additional calculations.
By rearranging or substituting within the point-slope formula, you can also discover various other forms or equations of the line, such as the slope-intercept form.

Line Through a Point

The phrase "line through a point" generally refers to a line drawn such that it passes through a specific fixed point in the coordinate plane.
If you know the point and the slope, you can easily write the line's equation using the point-slope form.
This concept is fundamental in geometry and algebra because any two unique points in the plane can define precisely one line.
To achieve this in practice when given a point \((x_1, y_1)\) and a slope \( m \), you can use the point-slope form mentioned previously.

• Start with your known point as your coordinates in the point-slope form equation.
• Substitute \( m \) for the slope.
• You'll end up with an equation specific to the line through this point and given slope.

In terms of parametric representation, envisioning the line as a path traced by a point moves with time can offer a different perspective on understanding the line's behavior.

Slope Parameterization

Parameterization introduces a parameter, often denoted by \( t \), that defines a line's position.
This approach allows for the representation of both the \( x \) and \( y \) coordinates separately, helping visualize the progression of points along the line.
For a line with slope \( m \) through the point \((x_1, y_1)\), a typical parametric form is:

• \( x = x_1 + t \)
• \( y = y_1 + mt \)

Parametric equations yield all the points on the line as the parameter \( t \) varies over real numbers.
Interestingly, you can alter the parameterization by substituting, for example, \( s = -t \), resulting in:

• \( x = x_1 - s \)
• \( y = y_1 - ms \)

Both parameterizations depict the same line but through different expressions.
They add versatility and are particularly useful in physics and computer graphics, where motion along a line could be explored as a continuous path as \( t \) or \( s \) evolves.