Problem 9 - 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. $$x=4 t^{2}, \quad y=8 t^{3}$$

Step-by-Step Solution

Answer

The curve is \(y = x^{3/2}\), resembling the right half of a parabola.

Step 1: Understand the Parametric Equations

The given parametric equations are \(x = 4t^2\) and \(y = 8t^3\). This means that both \(x\) and \(y\) are expressed in terms of the parameter \(t\). You are required to find the relationship between \(x\) and \(y\) by eliminating \(t\) and then sketch the curve using these equations.

Step 2: Solve for the Parameter

To eliminate \(t\), first solve one of the equations for \(t\). Start with \(x = 4t^2\). Solve for \(t^2\) by dividing both sides by 4: \(t^2 = \frac{x}{4}\). Now take the square root: \(t = \pm \sqrt{\frac{x}{4}}\).

Step 3: Substitute to Eliminate Parameter

Use the expression for \(t\) from Step 2 in the equation for \(y\). We have \(y = 8t^3\). Substitute \(t = \sqrt{\frac{x}{4}}\): \(y = 8\left(\sqrt{\frac{x}{4}}\right)^3 = 8 \cdot \left(\frac{x}{4}\right)^{3/2} = 8 \cdot \frac{x^{3/2}}{8} = x^{3/2}\).

Step 4: Write the Rectangular Equation

Now we have the rectangular-coordinate equation without any parameters: \(y = x^{3/2}\). This equation represents the relationship between \(x\) and \(y\) without using the parameter \(t\).

Step 5: Sketch the Curve

The equation \(y = x^{3/2}\) is a power function. Due to the context of the parametric equations, consider only the values of \(x\) and \(y\) resulting from non-negative \(t\), as \(t\) represents real numbers. The curve will start from the origin moving to the right and upward, resembling the right half of a parabola, due to \(x = 4t^2\) being always non-negative. Plot points for different \(t\) values such as 0, 1, and 2 to visualize it.

Key Concepts

Eliminating the Parameter

When working with parametric equations, the goal is often to convert them into a rectangular equation, which makes the relationship between the variables more direct. In the given exercise, the parametric equations are \(x = 4t^2\) and \(y = 8t^3\). Here, both \(x\) and \(y\) depend on the parameter \(t\). The process of eliminating the parameter involves expressing one variable in terms of the other, effectively removing \(t\) from the equations.

To start, we solve one of the parametric equations for the parameter. Here, solving \(x = 4t^2\) for \(t\) gives us \(t^2 = \frac{x}{4}\). By taking the square root, we have \(t = \pm \sqrt{\frac{x}{4}}\).

• It's important to note the plus-minus sign, which indicates two possible solutions for each \(x\).

Next, we substitute this expression for \(t\) into the second equation \(y = 8t^3\) to find \(y\) in terms of \(x\). Doing so, we reach our rectangular equation: \(y = x^{3/2}\). This transformation reveals the intrinsic relationship between \(x\) and \(y\) in a simplified form.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a way to graphically represent mathematical equations. It consists of two perpendicular axes: the horizontal \(x\)-axis and the vertical \(y\)-axis. Each point on the plane corresponds to a pair of coordinates \((x, y)\). This system facilitates the visualization of relationships between variables, particularly when they form a curve or a line.

Once we eliminate the parameter in a set of parametric equations, the resulting rectangular equation \(y = x^{3/2}\) fits neatly into this coordinate system. It allows us to see how changes in \(x\) correspond to changes in \(y\), without concerning ourselves with the intermediate parameter \(t\).

• It's helpful to understand this system as a map where each equation is a path traced through the plane.
• This approach makes it easier to comprehend how variables interact with each other visually.

In our exercise, the rectangular coordinate system acts as a canvas on which we can accurately sketch the curve described by the equation \(y = x^{3/2}\), translating algebra into geometry.

Sketching Curves

Sketching curves from equations is a valuable skill that helps visualize mathematical relationships. When sketching from parametric equations like \(x = 4t^2\) and \(y = 8t^3\), converting them to their rectangular form gives us the equation \(y = x^{3/2}\), which is easier to graph on a Cartesian plane.

To begin sketching, consider:

• Selecting points: Choose different values of \(t\) (such as 0, 1, 2) and calculate the corresponding points based on the parametric equations.
• Drawing the curve: Since we have the rectangular equation, plot these points on the \(xy\)-plane and connect them to form a curve. Note that the curve will only include points where \(x\geq 0\), as \(x = 4t^2\) is always non-negative.
• Observing symmetry: Given the nature of \(x = 4t^2\) and \(y = 8t^3\), the curve starts at the origin and extends upwards and to the right, resembling the right half of a parabola.

Through this process, you create a visual representation of the mathematical relationship between \(x\) and \(y\), enhancing your understanding of how the parametric equations define a path through the plane.