10. Parametric Equations

Writing Parametric Equations

10. Parametric Equations

Writing Parametric Equations: Videos & Practice Problems

Topic summary

To convert a rectangular equation into parametric equations, start by selecting a parameter, typically denoted as $t$ , which can be an expression involving $x$ or $y$ . Substitute this expression into the original equation to derive the corresponding $x$ and $y$ equations. Ensure to avoid domain restrictions by selecting odd powers of $x$ . This process allows for the creation of multiple valid parametric representations of the same geometric figure.

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Parameterizing Equations

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Parameterizing Equations Video Summary

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) $ yields $ y(t) = 2(t - 1) + 5 $, which simplifies to $ y(t) = 2t + 3 $.

Alternatively, if the equation is more complex, such as $ y = (x + 2)^2 - 3 $, a different choice for $ t $ might be necessary. Here, setting $ t = x + 2 $ allows for $ x(t) = t - 2 $. Substituting this into the equation results in $ y(t) = (t - 2)^2 - 3 $, which can be simplified to yield $ y(t) = t^2 - 3 $.

It is crucial to avoid domain restrictions when selecting $ t $. For example, using even powers of $ x $ or functions like square roots can lead to complications, as they may restrict the values $ t $ can take. Instead, opting for odd powers or linear transformations of $ x $ ensures that $ t $ can take on any real number, thus avoiding potential issues with imaginary numbers.

Finally, a useful verification step is to eliminate the parameter $ t $ from the resulting 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 $.

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Parameterizing Equations Example 1

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Parameterizing Equations Example 1 Video Summary

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 $. A valid option is to set $ t = x^3 $. This choice avoids complications with even powers, which can introduce domain restrictions. From this definition, we can solve for $ x $ in terms of $ t $: $ x(t) = \sqrt[3]{t} $. Substituting this back into the original equation yields:

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

Therefore, 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.

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Equations of Circles & Ellipses

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Equations of Circles & Ellipses Video Summary

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 like $\frac{x^2}{4} + \frac{y^2}{9} = 1$, you can recognize that it represents an ellipse. To parameterize this, you would first rewrite it as $x^2 + \frac{y^2}{9} = 1$ by multiplying through by 9. This allows you to set $f(x) = 2\cos(t)$ and $g(y) = 3\sin(t)$.

Next, isolate $x$ and $y$ to express them in terms of $t$. For $x$, you would have:

\[x = 2\cos(t)\]

And for $y$, you would derive:

\[y = 3\sin(t)\]

These equations, $x(t) = 2\cos(t)$ and $y(t) = 3\sin(t)$, are the parametric equations that describe the ellipse.

In another example, consider the equation $9x^2 + y^2 = 9$. To parameterize this, first divide the entire equation by 9 to obtain:

\[x^2 + \frac{y^2}{9} = 1\]

From here, you can set $x = \cos(t)$ and $y = 3\sin(t)$ after isolating the terms. This leads to the parametric equations:

\[x(t) = \cos(t)\]

\[y(t) = 3\sin(t)\]

These steps illustrate how to convert standard equations into parametric forms using trigonometric identities, allowing for a deeper understanding of the geometric representations of these equations.

Problem

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

$x=25\sin t$ ; $y=25\cos t$

$x=25\cos t$ ; $y=25\sin t$

$x=5\sin t$ ; $y=5\cos t$

$x=5\cos t$ ; $y=5\sin t$

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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 like $x + 1$ . 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 $x$ to prevent domain restrictions. For equations involving $x^{2} + y^{2}$ , use the Pythagorean identity to set $x$ and $y$ in terms of cosine and sine functions. Always check your parametric equations by eliminating the parameter to ensure you return to the original equation.

What is the process of parameterizing an equation?

Parameterizing an equation involves converting a rectangular equation into parametric form. Begin by selecting an expression for the parameter $t$ , typically involving $x$ or $y$ . Solve for $x (t)$ and substitute this into the original equation to find $y (t)$ . For equations like $x^{2} + y^{2}$ , use the Pythagorean identity to express $x$ and $y$ in terms of cosine and sine functions. Ensure the parametric equations describe the same relationship by checking them through the elimination of the parameter.

Why should you avoid even powers of x when choosing t?

When choosing an expression for the parameter $t$ , it is advisable to avoid even powers of $x$ such as $x^{2}$ or $\sqrt{x}$ . This is because solving for $x$ in terms of $t$ can lead to domain restrictions, resulting in imaginary numbers when $t$ is negative. Choosing odd powers or simple expressions involving $x$ helps avoid these issues, ensuring that $t$ can take any real number value without restrictions.

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 process involves solving one of the parametric equations for $t$ and substituting it into the other equation. If the resulting equation matches the original rectangular equation, your parametric equations are correct. This method serves as a reliable way to ensure that the parametric equations accurately describe the same relationship as the original equation.

What is the Pythagorean identity used in parameterizing equations?

The Pythagorean identity used in parameterizing equations is $\cos^{2} (t) + \sin^{2} (t) = 1$ . This identity is particularly useful for equations involving $x^{2} + y^{2}$ , as it allows you to express $x$ and $y$ in terms of cosine and sine functions of $t$ . By setting $x$ equal to $\cos (t)$ and $y$ equal to $\sin (t)$ , you can parameterize the equation, ensuring it describes a circle or ellipse.