Problem 6 - Precalculus: Mathematical for Calculus

Question

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=t^{2}, \quad y=t^{4}+1$$

Step-by-Step Solution

Answer

The curve is a parabola opening upwards: \(y = x^2 + 1\) for \(x \geq 0\).

Step 1: Understand the given parametric equations

The parametric equations provided are \(x = t^2\) and \(y = t^4 + 1\). Here, \(t\) is the parameter, \(x\) and \(y\) are given as functions of \(t\). These equations represent a curve in the plane as \(t\) varies.

Step 2: Sketch the curve from the parametric equations

To sketch the curve, choose various values of \(t\) both positive and negative, such as \(-2, -1, 0, 1, 2\), and calculate the corresponding \(x\) and \(y\) values. Plot these \((x, y)\) points on a Cartesian plane:- For \(t = -2\), \(x = 4\) and \(y = 17\)- For \(t = -1\), \(x = 1\) and \(y = 2\)- For \(t = 0\), \(x = 0\) and \(y = 1\)- For \(t = 1\), \(x = 1\) and \(y = 2\)- For \(t = 2\), \(x = 4\) and \(y = 17\)Connect these points to visualize the curve.

Step 3: Eliminate the parameter to find a rectangular equation

To eliminate the parameter \(t\), express \(t\) in terms of \(x\). From the equation \(x = t^2\), we have \(t = \pm \sqrt{x}\). Substitute into the second equation: if \(y = t^4 + 1\), then substituting for \(t^4\) yields \(y = (t^2)^2 + 1 = x^2 + 1\) for \(t = \sqrt{x}\). Thus, the rectangular equation is \(y = x^2 + 1\).

Step 4: Confirm the range of the rectangular equation

Since \(x = t^2\), \(x\) is always non-negative. Thus, the rectangular equation \(y = x^2 + 1\) defines a parabola opening upwards, starting at \(y = 1\) and for \(x \geq 0\). For all non-negative \(x\), the points lie above or on the parabola defined by \(y = x^2 + 1\).

Key Concepts

Curve Sketching

Curve sketching is a method of graphing the path traced by parametric equations as the parameter changes. In this exercise, we use the parametric equations \(x = t^2\) and \(y = t^4 + 1\). To graph the curve, it's essential to understand how \(x\) and \(y\) relate to different values of \(t\).Let's choose a few values of \(t\):

• For \(t = -2\), we get \((x, y) = (4, 17)\).
• For \(t = -1\), we get \((x, y) = (1, 2)\).
• For \(t = 0\), the point is \((x, y) = (0, 1)\).
• For \(t = 1\), we find \((x, y) = (1, 2)\).
• For \(t = 2\), the result is \((x, y) = (4, 17)\).

Plot these points on a graph and connect them to visualize the curve. You'll notice symmetry in the graph, which reflects through the vertical axis. The sketch gives insight into how the curve behaves over different intervals of \(t\). By observing these patterns, we get an intuitive idea about the overall behavior of the curve.

Rectangular-Coordinate Equation

The purpose of finding a rectangular-coordinate equation is to convert parametric equations into a form that involves only \(x\) and \(y\). This conversion removes the parameter \(t\), making it easier to analyze the curve in traditional terms.For the given parametric equations, we have \(x = t^2\) and \(y = t^4 + 1\). To eliminate \(t\), solve for \(t\) in terms of \(x\):\[t = \pm \sqrt{x}\]Substitute this into the second equation:\[y = (t^2)^2 + 1 = x^2 + 1\]This yields the rectangular equation \(y = x^2 + 1\). This equation represents a parabola that opens upwards. Removing the parameter allows us to better describe the curve through a well-known shape, making it simpler to predict its behavior.

Eliminating Parameters

Eliminating parameters involves expressing a parametric equation solely in terms of the variables \(x\) and \(y\), without the need for a parameter like \(t\). This step simplifies the representation of the curve.In this exercise, start with \(x = t^2\). Rewrite \(t\) as \(t = \pm \sqrt{x}\). Then substitute into \(y = t^4 + 1\). By performing this calculation:\[y = (\pm \sqrt{x})^4 + 1\ = x^2 + 1\]This step simplifies our parametric form to the rectangular form \(y = x^2 + 1\). One key point when eliminating parameters is to pay attention to allowable values of \(x\). Since \(t\) was squared to form \(x\), \(x\) must be non-negative (\(x \geq 0\)). Therefore, the rectangular equation only applies where \(x\) allows for real numbers under the square root transformation. This provides a clear and comprehensive understanding of the curve's domain and range.