Problem 2 - Precalculus

Question

You are given the parametric equations of a curve and a value for the parameter \(t\). Find the coordinates of the point on the curve corresponding to the given value of \(t\). $$x=3-t^{2}, y=4+t^{3} ; t=-1$$

Step-by-Step Solution

Answer

The point is (2, 3).

Step 1: Substitute the parameter into the x-equation

Given the parametric equations \(x = 3 - t^2\) and \(y = 4 + t^3\), we need to find the x-coordinate by substituting the given value \(t = -1\) into the equation for \(x\). \[x = 3 - (-1)^2 = 3 - 1 = 2\]

Step 2: Substitute the parameter into the y-equation

Next, we substitute \(t = -1\) into the equation for \(y\). \[y = 4 + (-1)^3 = 4 - 1 = 3\]

Step 3: Write the coordinates of the point

The coordinates of the point on the curve that correspond to \(t = -1\) are found by combining the results from Step 1 and Step 2. The point is \((x, y) = (2, 3)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics where we use algebra to study geometric problems. It allows us to describe geometric shapes using coordinate systems, usually the Cartesian coordinate system. Here, every point in the plane is described by a pair of numerical coordinates.

In the context of parametric equations, these coordinates are determined by substituting specific values into the parametric form. This involves equations for both the x-coordinate and the y-coordinate. This can be represented in a form where each point is described by

• x-value
• y-value

For example, with parametric equations such as those given, each value of the parameter 't' determines a unique point

• x-tuple being tied with a y-tuple

The substitution aspect here is key to revealing the actual location of the point in geometric terms.

Curve Plotting

When dealing with parametric equations, curve plotting involves more than just placing points on a graph. It involves understanding how these points, as determined by the parameter, unveil a complete shape or path as 't' varies.

To plot the curve given by the set of parametric equations such as

• \( x = 3 - t^2 \)
• \( y = 4 + t^3 \)

Different values of 't' are substituted, yielding different points. Plotting each of these points on the graph and connecting them reveals the overall shape of the curve.

The beauty of parametric equations is their ability to show continuous motion or paths, offering a different take from standard function graphs, which are usually in the form \( y = f(x) \). As 't' varies across its range, we can clearly see how a point "moves" along the curve, producing a pattern or shape. It’s like drawing by connecting the dots, only here, the dots are dynamically generated by the parametric equations.

Parameter Substitution

Parameter substitution is a crucial technique in solving parametric equations. This process involves replacing the parameter with a specific value, simplifying the computation and allowing us to determine exact coordinates on the curve.

In the exercise, we start with the parametric equations:

• \( x = 3 - t^2 \)
• \( y = 4 + t^3 \)

and substitute \( t = -1 \). By substituting into each equation separately

• \( x = 3 - (-1)^2 = 2 \)
• \( y = 4 + (-1)^3 = 3 \)

we find the specific point on the curve associated with this value of 't'.

This systematic approach enables us to not just find a single point, but potentially explore how different parameter values affect the position of points on the curve. It's a way of directly translating algebraic expressions into geometric coordinates for graph interpretation.