Writing Parametric Equations: Videos & Practice Problems

To convert a rectangular equation into parametric equations, start by defining a parameter \( t \) based on either \( x \) or \( y \). For example, if \( t = x + 1 \), solve for \( x(t) \) and substitute into the original equation to find \( y(t) \). This process can yield multiple valid parametric forms. Avoid using even powers of \( x \) to prevent domain restrictions. Always check your work by eliminating the parameter to ensure you return to the original equation. This method is essential for understanding relationships in equations involving circles or ellipses.

Parameterizing Equations

In the process of converting a rectangular equation into parametric equations, the first step involves selecting a suitable expression for the parameter \( t \). This expression can be based on either \( x \) or \( y \). A common approach is to set \( t \) equal to \( x \) or a simple transformation of \( x \), such as \( t = x + 1 \). This choice allows for straightforward calculations, as it simplifies the relationship between the variables.

Once \( t \) is defined, the next step is to express \( x \) in terms of \( t \). For instance, if \( t = x + 1 \), then rearranging gives \( x(t) = t - 1 \). This expression can then be substituted back into the original equation to find \( y(t) \). For example, if the original equation is \( y = 2x + 5 \), substituting \( x(t) \) results in \( y(t) = 2(t - 1) + 5 \), which simplifies to \( y(t) = 2t + 3 \).

In cases where the equation is more complex, such as \( y = (x + 2)^2 - 3 \), a similar method applies. If we choose \( t = x + 2 \), then \( x(t) = t - 2 \). Substituting this into the equation yields \( y(t) = (t - 2)^2 - 3 \). After simplifying, this results in \( y(t) = t^2 - 3 \).

It is important to avoid choosing \( t \) in a way that introduces domain restrictions, such as using even powers of \( x \) or square roots, which can lead to complex numbers when solving for \( x \). Instead, selecting odd powers or simple transformations of \( x \) ensures that \( t \) can take on any real number value.

Finally, a useful verification step is to eliminate the parameter \( t \) from the parametric equations to check if the original rectangular equation is recovered. This confirms that the chosen parametric equations accurately represent the same relationship between \( x \) and \( y \).

Parameterizing Equations Example 1

In this discussion, we explore how to convert the rectangular equation \( y = 2x^3 \) into parametric equations. The process begins by defining a parameter \( t \). A common approach is to set \( t = x \), which simplifies the parameterization. In this case, we can express \( x \) in terms of \( t \) as \( x(t) = t \). Substituting this into the original equation gives us the corresponding \( y \) equation: \( y(t) = 2(t^3) \). Thus, the first set of parametric equations is:

\( x(t) = t \)

\( y(t) = 2t^3 \)

Alternatively, we can choose a different definition for \( t \). For instance, setting \( t = x^3 \) is another valid option. This leads us to express \( x \) in terms of \( t \) as \( x(t) = \sqrt[3]{t} \). Substituting this back into the original equation results in:

\( y(t) = 2(\sqrt[3]{t})^3 = 2t \)

Thus, the second set of parametric equations is:

\( x(t) = \sqrt[3]{t} \)

\( y(t) = 2t \)

Both sets of parametric equations effectively describe the same relationship as the original rectangular equation, demonstrating the flexibility in parameterization methods. Understanding these transformations is crucial for analyzing curves and their properties in calculus and analytical geometry.

Equations of Circles & Ellipses

In the study of parametric equations, a common task is to convert equations involving \(x\) and \(y\) into parametric forms. This process often utilizes the Pythagorean identity, which states that \(\cos^2(t) + \sin^2(t) = 1\). When given an equation like \(x^2 + y^2 = 1\), it can be parameterized by expressing \(x\) and \(y\) in terms of a parameter \(t\).

To begin, identify if the equation can be rewritten in the form \(f(x)^2 + g(y)^2 = 1\). For example, if you have an equation such as \(\frac{x^2}{4} + \frac{y^2}{9} = 1\), you can recognize that it resembles the standard form of an ellipse. To parameterize this, you would set \(f(x)\) to \(\frac{x}{2}\) and \(g(y)\) to \(\frac{y}{3}\), leading to the equations:

\(x = 2\cos(t)\) and \(y = 3\sin(t)\).

Next, if the equation is not already in this form, you can manipulate it by dividing through by the constant on the right side. For instance, in the equation \(9x^2 + y^2 = 9\), dividing by 9 gives:

\(x^2 + \frac{y^2}{9} = 1\).

From here, you can set \(x^2\) equal to \(\cos^2(t)\) and \(\frac{y^2}{9}\) equal to \(\sin^2(t)\). Solving for \(x\) and \(y\) yields:

\(x = 3\cos(t)\) and \(y = 3\sin(t)\).

This method effectively allows you to derive parametric equations that describe the same geometric figure represented by the original equation. Always remember that you can verify your parameterization by eliminating the parameter \(t\) to return to the original equation.

In summary, the key steps to parameterizing an equation are to rewrite it in the standard form, apply the Pythagorean identity, and solve for \(x\) and \(y\) in terms of the parameter \(t\). This approach not only simplifies the process but also reinforces the relationship between trigonometric functions and geometric shapes.

Write parametric equations for the rectangular equation below.
$x^2+y^2=25$

• A. $x=25\sin t$; $y=25\cos t$
• B. $x=25\cos t$; $y=25\sin t$
• C. $x=5\sin t$; $y=5\cos t$
• D. $x=5\cos t$; $y=5\sin t$

Answer: (D)

Here’s what students ask on this topic:

How do you write parametric equations from a rectangular equation?

To write parametric equations from a rectangular equation, start by choosing an expression for the parameter $t$, often involving $x$ or $y$. A common choice is to set $t$ equal to $x$ or a simple expression involving $x$, such as $x+\; 1$. After defining $t$, solve for $x(t)$ and substitute this into the original equation to find $y(t)$. Avoid choosing $t$ as even powers of $x$ (like $x^2$) to prevent domain restrictions, as this can lead to imaginary numbers. Always check your work by eliminating the parameter to ensure you return to the original equation.

What is the process for parameterizing equations involving $x^2+y^2?$

For equations involving $x^2+y^2,\; use\; the\; Pythagorean\; identity\; to\; set$ x$and$ y$in\; terms\; of\; cosine\; and\; sine\; functions.\; For\; example,\; if\; you\; have\; an\; equation\; like$ x^2+y^2=\; 1$,\; you\; can\; set$ x(t)\; =\; cos(t)$and$ y(t)\; =\; sin(t)$.\; This\; parameterization\; works\; because$ t^2+t^2=\; 1$by\; the\; Pythagorean\; identity.\; This\; method\; is\; particularly\; useful\; for\; circles\; and\; ellipses.$

Why should you avoid choosing $x^2$ as a parameter in parametric equations?

When writing parametric equations, it's advisable to avoid choosing $x^2$ as a parameter because it can lead to domain restrictions. Specifically, if you solve for $x$ in terms of $t$ and end up with a square root, $t$ cannot be negative, as this would result in imaginary numbers. This restriction limits the range of values $t$ can take, which can complicate the parameterization process. Instead, choose $t$ as a linear expression involving $x$, such as $x$ or $x+\; 1$, to avoid these issues.

How can you check if your parametric equations are correct?

To verify the correctness of your parametric equations, you can eliminate the parameter $t$ and check if you return to the original rectangular equation. This involves solving one of the parametric equations for $t$ and substituting it into the other equation to eliminate $t$. If the resulting equation matches the original rectangular equation, your parametric equations are correct. This process is essentially the reverse of parameterization and serves as a reliable method to ensure accuracy.

What are some common choices for the parameter $t$ in parametric equations?

Common choices for the parameter $t$ in parametric equations include setting $t$ equal to $x$ or a simple expression involving $x$, such as $x+\; 1$. These choices are straightforward and help avoid domain restrictions. For equations involving trigonometric identities, $t$ can be set to angles in cosine and sine functions, such as $x(t)\; =\; cos(t)$ and $y(t)\; =\; sin(t)$. These choices simplify the parameterization process and ensure that the resulting parametric equations are valid over a wide range of values.