Eliminating the Parameter t to Determine the Cartesian Equation

In mathematics, parametric equations are a way to represent a curve or a set of points using a parameter, often denoted as 't'. These equations can be useful in various applications, such as describing the motion of an object or representing complex shapes. However, sometimes, it is necessary to express the relationship between the variables in a Cartesian form, which is a direct equation involving the x and y coordinates.

In this article, we will explore the process of eliminating the parameter 't' to determine the Cartesian equation for the given parametric equations: x = t^2 and y = 8 + 4t.

Step 1: Isolate the Parameter t

To begin, we need to isolate the parameter 't' from one of the given equations. In this case, let's use the equation y = 8 + 4t:

y = 8 + 4t
t = (y - 8) / 4

Step 2: Substitute the Expression for t into the Other Equation

Now that we have an expression for 't' in terms of 'y', we can substitute it into the other equation, x = t^2:

x = t^2
x = ((y - 8) / 4)^2

Step 3: Simplify the Expression

To simplify the expression, we can expand the squared term and rearrange the equation:

x = ((y - 8) / 4)^2
x = (y^2 - 16y + 64) / 16
x = (y^2 - 16y + 64) / 16

Step 4: Determine the Cartesian Equation

The final step is to express the relationship between 'x' and 'y' in a Cartesian form. We can do this by rearranging the equation to isolate 'y':

x = (y^2 - 16y + 64) / 16
16x = y^2 - 16y + 64
y^2 - 16y + 64 - 16x = 0

Therefore, the Cartesian equation for the given parametric equations is:

y^2 - 16y + 64 - 16x = 0

This equation represents the relationship between the 'x' and 'y' coordinates, without the need for the parameter 't'.

By following these steps, you can eliminate the parameter 't' and determine the Cartesian equation for any set of parametric equations. This technique is essential for high school mathematics students and teachers, as it allows them to work with different representations of curves and shapes, and to understand the connections between them.