In Exercises 53–56, find two different sets of parametric equations for each rectangular equation. y = 4x − 3

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a variable, often denoted as 't'. For example, in the context of the equation y = 4x - 3, we can define x and y in terms of a parameter t, such as x = t and y = 4t - 3. This allows for a more flexible representation of curves, especially in cases where a single variable equation may not capture the full behavior of the graph.

Rectangular Equations

A rectangular equation relates the x and y coordinates of points in a Cartesian plane without involving a parameter. The equation y = 4x - 3 is a linear equation representing a straight line with a slope of 4 and a y-intercept of -3. Understanding how to convert between rectangular and parametric forms is essential for analyzing the same geometric relationship from different perspectives.

Slope-Intercept Form

The slope-intercept form of a linear equation is given by y = mx + b, where m represents the slope and b the y-intercept. In the equation y = 4x - 3, the slope is 4, indicating that for every unit increase in x, y increases by 4 units. Recognizing this form helps in deriving parametric equations that maintain the same slope and intercept, ensuring the parametric representation accurately reflects the original linear relationship.