Problem 26 - A Graphical Approach to College Algebra

Question

For each plane curve, find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=\sqrt{t}, y=t^{2}-1, \text { for } t \text { in }[0, \infty)$$

Step-by-Step Solution

Answer

The rectangular equation is \(y = x^4 - 1\) with \(x \geq 0\).

Step 1: Express t in terms of x

We start by solving the equation \(x = \sqrt{t}\) for \(t\). By squaring both sides, we get \(t = x^2\).

Step 2: Substitute t in y equation

Next, take the expression we found for \(t\) from Step 1 and substitute it into \(y = t^2 - 1\). So, \(y = (x^2)^2 - 1\).

Step 3: Simplify the y equation

Simplify the expression for \(y\) from Step 2: \(y = x^4 - 1\). Now, we have the rectangular equation \(y = x^4 - 1\).

Step 4: Determine the interval for x

Since \(t = \sqrt{x}\) and \(t\) is defined for \([0, \infty)\), \(t\) is non-negative. Therefore, \(x = \sqrt{t}\) implies \(x \geq 0\). The appropriate interval for \(x\) is \([0, \infty)\).

Key Concepts

Plane Curve

A plane curve is a smooth, one-dimensional line that lies within a two-dimensional plane. It’s typically represented using parametric equations, which employ a parameter like \(t\) to express coordinates \((x, y)\) in the plane. For example, the parametric equations given are \(x = \sqrt{t}\) and \(y = t^2 - 1\). Here, as \(t\) changes, the values of \(x\) and \(y\) change, tracing a path on the plane. Understanding plane curves is essential because it helps visualize shapes and paths and analyze their properties.
Switching from parametric to rectangular equations involves eliminating the parameter \(t\), as seen in the exercise. This conversion simplifies the representation and aids in analyzing intersections and symmetries more easily.
Overall, plane curves provide a geometric understanding of mathematical functions and relations.

Interval for x

Finding the interval for \(x\) is about identifying the range of values \(x\) can take based on our equations and conditions. Given \(x = \sqrt{t}\), and knowing \(t\) is defined on \([0, \infty)\), \(t\) must be non-negative. Consequently, since the square root function is only defined for non-negative numbers, \(x\) is also non-negative; thus, \(x \geq 0\).
The appropriate interval for \(x\) is \([0, \infty)\), meaning \(x\) can be any non-negative number, reaching indefinitely towards positive infinity.
This interval is crucial as it defines the domain where the derived rectangular equation \(y = x^4 - 1\) is valid in terms of \(x\). Understanding this helps in graphing the equation and in predicting its behavior within that specific set of possible values for \(x\).

Solving Equations

Solving equations involves manipulating an equation to find values for the unknown variable that will satisfy it. Here, the key is transitioning from a parametric form to a rectangular form. We start by isolating the parameter. The equation \(x = \sqrt{t}\) is solved for \(t\) by squaring both sides to obtain \(t = x^2\).
Once we have an expression for \(t\) in terms of \(x\), substitute this back into the other equation: \(y = t^2 - 1\), resulting in the rectangular form \(y = (x^2)^2 - 1\). After simplifying this, we get \(y = x^4 - 1\).
This method allows us to eliminate the parameter, directly relating \(x\) to \(y\). Solving equations in this manner is critical for simplifying complex relationships and allows easier analysis of their behavior and properties when graphed.