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.


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