Problem 25 - Contemporary Precalculus

Question

The given curve is part of the graph of an equation in \(x\) and \(y .\) Find the equation by eliminating the parameter. $$x=\sqrt{t}, \quad y=t^{4}-1, \quad t \geq 0$$

Step-by-Step Solution

Answer

Question: Eliminate the parameter \(t\) from the parametric equations \(x = \sqrt{t}\) and \(y = t^4 - 1\) to obtain the equation of the curve in terms of \(x\) and \(y\). Answer: \(y = x^8 - 1\)

Step 1: Solve the first equation for \(t\)

From the first equation, we have \(x = \sqrt{t}\). We can square both sides to eliminate the square root, giving: \(x^2 = t\).

Step 2: Substitute \(t\) into the second equation

Now we have an expression for \(t\) from step 1, which is \(x^2\). We can substitute this expression into the second equation to eliminate \(t\). So, the second equation becomes: \(y = (x^2)^4 - 1 = x^8 - 1\).

Step 3: Final expression for the curve

Combining steps 1 and 2, we have the following relationship between \(x\) and \(y\): \(y = x^8 - 1\). This is the equation of the curve without the parameter \(t\).

Key Concepts

Parameter Elimination

The technique of parameter elimination is used to convert a set of parametric equations into a single equation that directly relates the variables involved, normally without the parameter. This is particularly useful when we want to understand the relationship between the variables without the need for a third variable, or when we need to graph the set of equations on a standard Cartesian plane.

In the exercise provided, the parameter 't' is eliminated through a two-step process. First, one of the equations is solved for 't' and then 't' is substituted back into the other equation. This substitution provides a direct relation between 'x' and 'y', effectively removing the parameter 't'. When eliminating parameters, it's essential to consider the domain or range of the original parametric equations, as the elimination may introduce solutions that weren't present in the original parametric form.

Parametric Equations

Parametric equations are a set of equations where each coordinate dimension is expressed in terms of a single variable called a parameter. Instead of describing a curve by an equation in 'x' and 'y', parametric equations use a third variable 't', which represents a quantity like time, to specify points on the curve. This parameter 't' can provide additional context and allow the representation of more complex curves that cannot be easily defined using only 'x' and 'y'.

The given exercise showcases parametric equations where the position on a curve is defined by \( x = \sqrt{t} \) and \( y = t^4 - 1 \) with 't' being a non-negative parameter. Here, 'x' and 'y' are both functions of 't', with 't' essentially 'driving' the motion along the curve.

Solving Parametric Equations

Solving parametric equations involves finding an algebraic relationship between the variables without the parameter—essentially, parameter elimination. The challenge often lies in expressing one variable solely in terms of the other, which sometimes requires algebraic manipulation like factoring, squaring, or solving for the parameter.

To make the concept clearer, let's consider the provided exercise. The first step is to express 't' in terms of 'x', which is done by squaring \( x = \sqrt{t} \) to obtain \( t = x^2 \). In the second step, this expression for 't' is substituted into the second equation to express 'y' solely in terms of 'x'. The result, \( y = x^8 - 1 \) establishes the direct relationship expected from solving parametric equations and is the final form of the curve, showing that for any value of 'x', we can now directly compute a corresponding value of 'y' without needing the parameter 't'.