# Consider the parametric equations below x t2 1 y t 4 3 t 3

## Introduction

These parametric equations define a curve in the xy-plane. The parametric equations are defined by two functions, x(t) and y(t). The function x(t) defines the x-coordinate of a point on the curve as a function of t, and the function y(t) defines the y-coordinate of the point on the curve as a function of t.

## The parametric equations

The parametric equations are a set of equations that express the coordinates of a point in terms of a parameter. In this case, the parameter is t and the coordinates are x and y. The equations are as follows:

x = t2 – 1
y = t + 4
3t = 3

These equations define a curve in the xy-plane. The value of t determines the position of the point on the curve. For example, if t = 0, then x = -1 and y = 4. If t = 1, then x = 0 and y = 5. If t = 2, then x = 3 and y = 6.

## The domain and range

The domain of a function is the set of all input values for which the function produces a result. The range of a function is the set of all output values for which the function produces a result.

For the parametric equations given above, the domain is all real numbers, and the range is all real numbers.

## The graph of the parametric equations

The graph of the parametric equations will be a parabola. The parabola will open upward if a > 0 and downward if a < 0. The vertex of the parabola will be at (1, 4/3).

## Conclusion

After examining the equation, it is clear that as t increases, so does x and y. However, x increases at a faster rate than y. Therefore, it can be concluded that the parametric equations represent a line that is slanted upwards and to the right.