Problem 9 - Precalculus with Limits

Question

In Exercises 7-26, (a) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation if necessary. \(x=\frac{1}{4}t\) \(y=t^2\)

Step-by-Step Solution

Answer

The rectangular equation that represents the curve is \(y=16x^2\). The curve moves upwards and rightwards as the parameter \(t\) increases.

Step 1: Sketch the parametric equations

First plot the graph of \(x=\frac{1}{4}t\) and \(y=t^2\). As the parameter \(t\) increases, both \(x\) and \(y\) increase. This indicates that the curve moves upwards along the y-direction and rightwards along the x-direction as \(t\) increases.

Step 2: Eliminate the parameter and find the rectangular equation

To eliminate the parameter \(t\), solve the equation \(x=\frac{1}{4}t\) for \(t\) to get \(t=4x\). Then substitute this in the \(y\) equation to get \(y=(4x)^2\), which simplifies to \(y=16x^2\).

Step 3: Adjust the domain of the resulting rectangular equation if necessary

Since \(x=\frac{1}{4}t\) can take all real values depending on the value of \(t\) and there is no limitation on the domain of \(t\), no adjustment for the domain is necessary for the rectangular equation.

Key Concepts

Rectangular Equation

In mathematics, converting parametric equations to a rectangular equation is a common way to represent a curve without extra parameters. Parametric equations use an independent parameter, often denoted as \( t \), to define both \( x \) and \( y \). To find the rectangular equation, we eliminate this parameter.

In the given example, the parametric equations are \( x = \frac{1}{4}t \) and \( y = t^2 \). To eliminate \( t \), solve for \( t \) in terms of \( x \): \( t = 4x \). This expression for \( t \) is then substituted into the equation for \( y \), yielding \( y = (4x)^2 = 16x^2 \). The resulting rectangular equation, \( y = 16x^2 \), represents the same curve as the original parametric form, but without the parameter \( t \).

Converting to a rectangular equation can often simplify analysis and graphing.

Curve Orientation

Curve orientation describes how a curve progresses as the parameter varies. For parametric equations, understanding the orientation is key to interpreting the motion along a curve.

In this exercise, as the parameter \( t \) increases, both \( x \) and \( y \) increase, indicating that the curve moves upwards and to the right. This is important because orientation tells us the direction of motion, which is not immediately visible in the rectangular equation.

• The absence of negative values for \( y = t^2 \) suggests all points lie in the non-negative \( y \)-side.
• The rightward movement is indicated as \( x \) increases progressively with increasing \( t \).

Recognizing the orientation helps when sketching the curve, ensuring directionality is accurately reflected.

Domain Adjustment

When translating parametric equations to rectangular forms, the discussion of domain is crucial. The domain refers to the set of possible \( x \)-values that make sense in the context of the equations.

For this specific problem, after elimination of the parameter, the rectangular equation is \( y = 16x^2 \). Originally, in parametric form, there are no constraints on \( t \) imposed by either \( x = \frac{1}{4}t \) or \( y = t^2 \), making \( t \) unrestricted. Consequently, \( x \) also embraces all real numbers. This differs from certain cases where the range of \( t \) might limit \( x \) and \( y \) values.

The domain of the rectangular equation remains unrestricted in this example, illustrating no modification is needed. When translating equations, always inspect whether the full range of \( t \) maps cleanly into the \( x \)-domain without extra conditions. This ensures faithfully representing the original curve.