Eliminate the parameter to rewrite the following as a rectangular equation. x ( t ) = 2 t − 1 x\left(t\right)=2t-1 x ( t ) = 2 t − 1 y ( t ) = t 5 − 2 y\left(t\right)=t^5-2 y ( t ) = t 5 − 2

y = ( x + 1 2 ) 5 − 2 y=\left(\frac{x+1}{2}\right)^5-2 y = ( 2 x + 1 ) 5 − 2

y = ( x − 2 2 ) 5 − 1 y=\left(\frac{x-2}{2}\right)^5-1 y = ( 2 x − 2 ) 5 − 1

y = ( x − 1 2 ) 5 − 2 y=\left(\frac{x-1}{2}\right)^5-2 y = ( 2 x − 1 ) 5 − 2

y = ( x + 1 2 ) 5 − 1 y=\left(\frac{x+1}{2}\right)^5-1 y = ( 2 x + 1 ) 5 − 1

First eliminate the parameter, then graph the plane curve of the parametric equations. x ( t ) = 2 + cos ⁡ t x\left(t\right)=2+\cos t x ( t ) = 2 + cos t , y ( t ) = − 1 + sin ⁡ t y\left(t)=-1+\sin t\right. ; 0 ≤ t ≤ 2 π 0\le t\le2\pi 0 ≤ t ≤ 2 π

( x − 2 ) 2 + ( y + 1 ) 2 = 1 \left(x-2\right)^2+\left(y+1\right)^2=1 ( x − 2 ) 2 + ( y + 1 ) 2 = 1

( x + 2 ) 2 + ( y − 1 ) 2 = 1 \left(x+2\right)^2+\left(y-1\right)^2=1 ( x + 2 ) 2 + ( y − 1 ) 2 = 1

What are parametric equations and how do you graph them?

Parametric equations express two variables, x and y , in terms of a third variable, typically denoted as t , called the parameter. These equations are written as x(t) and y(t) . To graph parametric equations, follow these steps: Create a table of values with three columns: t , x(t) , and y(t) . Choose or use given values for t . Substitute each t value into the x(t) and y(t) equations to calculate corresponding x and y values. Plot the resulting ( x , y ) points on a coordinate plane. Connect the points in the order of increasing t values, adding arrows to indicate direction. This process helps visualize the curve or line described by the parametric equations.

How do you eliminate the parameter in parametric equations?

To eliminate the parameter t in parametric equations, follow these steps: Solve one of the parametric equations (either x(t) or y(t) ) for t . Substitute the expression for t into the other equation. Simplify the resulting equation to express y in terms of x (or vice versa). For example, if x(t) = t + 1 and y(t) = t 2 - 3 , solve x(t) for t : t = x - 1 . Substitute this into y(t) : y = ( x - 1 ) 2 - 3 . The resulting equation, y = x 2 - 2 x - 2 , is the rectangular form.

How do you parameterize a rectangular equation?

To parameterize a rectangular equation, follow these steps: Choose an expression for t , often setting t = x or t as a simple function of x or y . Express x and y in terms of t . For example, if y = 2x + 3 , set t = x . Then, x(t) = t and y(t) = 2t + 3 . Ensure the parameterization avoids domain restrictions. For instance, avoid square roots or even powers of x that could restrict t to non-negative values. For more complex equations, such as y = (x + 2) 2 - 3 , you might set t = x + 2 , leading to x(t) = t - 2 and y(t) = t 2 - 3 . This process is the reverse of eliminating the parameter.

What are some common applications of parametric equations?

Parametric equations are widely used in various fields due to their ability to describe motion and curves. Common applications include: Physics: Modeling projectile motion, where x(t) and y(t) describe horizontal and vertical positions over time. Engineering: Designing paths for robotic arms or vehicles, where parametric equations define precise trajectories. Computer Graphics: Creating curves and animations, such as Bézier curves for smooth transitions. Mathematics: Representing complex curves like ellipses, circles, and parabolas. Geography: Mapping paths on Earth's surface, such as great-circle routes. These equations are powerful tools for describing dynamic systems and geometric shapes in a flexible and intuitive way.

16. Parametric Equations & Polar Coordinates

Parametric Equations

16. Parametric Equations & Polar Coordinates

Parametric Equations: Videos & Practice Problems

Learn Concepts Practice Worksheet

Topic summary

Parametric equations express two variables, x and y, in terms of a third variable, t. To graph these equations, create a table of values for t, x(t), and y(t), then plot the corresponding points. Eliminating the parameter involves substituting one equation into another to derive a rectangular equation. For trigonometric functions, use Pythagorean identities to relate sine and cosine. When parameterizing a rectangular equation, choose a simple expression for t, ensuring it avoids domain restrictions, and substitute back to find x(t) and y(t).

concept

Introduction to Parametric Equations

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Introduction to Parametric Equations Video Summary

In mathematics, parametric equations are a way to express two variables, typically $x$ and $y$, in terms of a third variable known as the parameter, usually denoted as $t$. This means that instead of having a single equation like $y = 2x - 3$, you will have two equations: $x(t)$ and $y(t)$. Understanding how to graph these equations is essential, as it allows for the representation of more complex relationships between the variables.

To graph parametric equations, the process is similar to graphing standard equations. You will create a table of values that includes three columns: one for the parameter $t$, one for $x$, and one for $y$. The values of $t$ can either be provided or chosen, and you will substitute these values into the equations to find the corresponding $x$ and $y$ values. For example, if you have the equations $x(t) = t + 1$ and $y(t) = 2t - 1$, you can calculate the points by plugging in values for $t$. When $t = 1$, $x = 2$ and $y = 1$; when $t = 2$, $x = 3$ and $y = 3$; and so on.

Once you have a set of points, you can plot them on a coordinate plane. It’s important to note that the graph of parametric equations is referred to as a plane curve, which can take various forms, including lines, parabolas, or more complex shapes. Unlike standard equations, parametric equations do not have a defined axis for $t$, so it is common to indicate the corresponding $t$ values next to their coordinates. This helps clarify the direction of the graph, as the orientation of the curve is determined by the increasing values of $t$.

To indicate the direction of the graph, arrows are often drawn along the curve, showing how the graph progresses as $t$ increases. This is a crucial aspect of parametric equations, as it provides insight into the behavior of the graph over time or along the parameter's range.

In summary, parametric equations allow for a flexible representation of relationships between variables, and understanding how to graph them involves creating a table of values, plotting points, and indicating the direction of the curve. This foundational knowledge is essential for exploring more advanced topics in mathematics.

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Problem

Graph the plane curve formed by the parametric equations and indicate its orientation.
$x (t) = - t + 1$ ; $y (t) = t^{2}$
$negative 2 is less than or equal to t is less than or equal to 2$

Graph of parametric equations with arrows indicating orientation.

Problem

Graph the plane curve formed by the parametric equations and indicate its orientation.
$x (t) = 2 t - 1$ ; $y (t) = 2 \sqrt{t}$
$t is greater than or equal to 0$

Graph of a parametric curve with arrows indicating direction, showing points plotted on a grid.

Graph of a linear parametric curve with arrows indicating direction, showing points plotted on a grid.

concept

Eliminating the Parameter

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Eliminating the Parameter Video Summary

In the study of parametric equations, the process of eliminating the parameter is essential for converting parametric forms into rectangular equations, which only involve the variables x and y. This transformation allows for a clearer understanding of the relationship between the two variables. To eliminate the parameter, one typically solves one of the parametric equations for the parameter (often denoted as t) and substitutes this expression into the other equation.

For instance, consider the parametric equations given by $ x(t) = \sqrt{t} $ and $ y(t) = t - 3 $. To eliminate the parameter, we can first express $ t $ in terms of $ x $ by squaring both sides of the equation for $ x $: $ t = x^2 $. Next, we substitute this expression for $ t $ into the equation for $ y $: $ y = t - 3 $ becomes $ y = x^2 - 3 $. This results in the rectangular equation $ y = x^2 - 3 $, which is a parabola that opens upwards.

It is important to note that the range of t values can affect the portion of the graph that is represented. In this case, since $ t $ must be non-negative (as indicated by the square root), the resulting graph of the parabola is only the upper half, starting from the vertex at (0, -3) and extending to the right. This illustrates how restrictions on the parameter can limit the graph to a specific segment of the overall shape defined by the rectangular equation.

When graphing parametric equations, it is also useful to calculate specific points by substituting values for t. For example, if $ t = 0 $, then $ x = 0 $ and $ y = -3 $. If $ t = 1 $, then $ x = 1 $ and $ y = -2 $. Continuing this process for other values of t helps to visualize the curve, confirming that it resembles a parabola.

In summary, eliminating the parameter from parametric equations involves substituting expressions derived from one equation into another, leading to a rectangular equation that can be graphed. Understanding this process not only aids in graphing but also enhances comprehension of the underlying relationships between the variables involved.

Problem

Eliminate the parameter to rewrite the following as a rectangular equation.
$x\left(t\right)=2t-1$
$y\left(t\right)=t^5-2$

$y=\left(\frac{x-2}{2}\right)^5-1$

$y=\left(\frac{x-1}{2}\right)^5-2$

$y=\left(\frac{x+1}{2}\right)^5-2$

$y=\left(\frac{x+1}{2}\right)^5-1$

example

Eliminating the Parameter Example 1

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Eliminating the Parameter Example 1 Video Summary

In this example, we explore the process of graphing parametric equations and converting them into their equivalent rectangular form. The goal is to eliminate the parameter, resulting in an equation that solely involves the variables x and y.

To begin graphing, we first identify the parametric equations, which are quadratic in nature, featuring the variable t. Given that the interval for t encompasses all real numbers, we select small values for t, such as 0, 1, 2, and 3, to generate corresponding x and y coordinates. This choice simplifies calculations, especially since both equations involve t squared, leading to symmetrical outputs for positive and negative values of t.

For the x-coordinate, we compute:

When t = 0: $ x = 0^2 - 1 = -1 $
When t = 1: $ x = 1^2 - 1 = 0 $
When t = 2: $ x = 2^2 - 1 = 3 $
When t = 3: $ x = 3^2 - 1 = 8 $

For the y-coordinate, we calculate:

When t = 0: $ y = 0^2 - 2 = -2 $
When t = 1: $ y = 1^2 - 2 = -1 $
When t = 2: $ y = 2^2 - 2 = 2 $
When t = 3: $ y = 3^2 - 2 = 7 $

These calculations yield the points (-1, -2), (0, -1), (3, 2), and (8, 7). Upon plotting these points, one might initially expect a parabolic shape due to the quadratic nature of the equations. However, connecting the points reveals a straight line, indicating that the graph does not represent a parabola but rather a linear relationship.

To determine the orientation of the graph, we can label the points with their corresponding t values, showing that the line progresses in the direction of increasing t.

Next, we derive the equivalent rectangular equation by eliminating the parameter t. Starting with the x equation, we express t in terms of x:

$ x = t^2 - 1 $ leads to $ t^2 = x + 1 $ and thus $ t = \sqrt{x + 1} $.

Substituting this expression for t into the y equation gives:

$ y = t^2 - 2 = \sqrt{x + 1}^2 - 2 $.

Since squaring the square root cancels out, we simplify this to:

$ y = (x + 1) - 2 = x - 1 $.

This final equation, $ y = x - 1 $, confirms that the relationship between x and y is linear, with a slope of 1 and a y-intercept of -1. This outcome illustrates that even when starting with quadratic parametric equations, the resulting graph can be a simple linear equation.

example

Eliminating the Parameter Example 2

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Eliminating the Parameter Example 2 Video Summary

In this discussion, we explore parametric equations defined as $ x(t) = t - 3 $ and $ y(t) = \frac{1}{t - 5} $. The goal is to graph the corresponding plane curve and derive the equivalent rectangular equation. A key point to note is that $ t $ can take any value except $ 5 $, as this would make the denominator in the $ y(t) $ equation zero, resulting in an undefined expression.

To graph the curve, we substitute given $ t $ values into the equations to obtain $ (x, y) $ coordinate pairs. For instance, when $ t = 0 $, $ x(0) = -3 $ and $ y(0) = -0.2 $. Continuing this process for other $ t $ values, we find coordinates such as $ (0, -0.5) $, $ (1.8, -5) $, $ (2.2, 5) $, and $ (4, 0.5) $. Plotting these points reveals a curve that approaches a vertical asymptote at $ x = 2 $ as $ t $ approaches $ 5 $ from either side, indicating the behavior of a rational function.

Next, we derive the rectangular equation by eliminating the parameter $ t $. Starting with the equation for $ x $, we rearrange it to find $ t = x + 3 $. Substituting this expression into the $ y(t) $ equation gives us:

\[y = \frac{1}{(x + 3) - 5} = \frac{1}{x - 2}\]

This resulting equation, $ y = \frac{1}{x - 2} $, confirms that we have a rational function with a vertical asymptote at $ x = 2 $. This behavior aligns with our expectations based on the original parametric equations.

In summary, we have successfully graphed the parametric equations and derived the equivalent rectangular equation, illustrating the relationship between the two forms and the characteristics of the resulting rational function.

concept

Eliminate Parameter: Equations with Trig

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Eliminate Parameter: Equations with Trig Video Summary

In the study of parametric equations, particularly those involving trigonometric functions, a systematic approach is essential for eliminating the parameter. When given equations such as $ x = \cos(t) $ and $ y = 3\sin(t) $, the first step is to isolate the trigonometric functions. For instance, from $ x = \cos(t) $, we can express this as $ \cos(t) = x $. Similarly, from $ y = 3\sin(t) $, we can rearrange it to find $ \sin(t) = \frac{y}{3} $.

To eliminate the parameter $ t $, we can utilize the Pythagorean identity, which states that $ \sin^2(t) + \cos^2(t) = 1 $. This identity is applicable regardless of the angle, meaning we can substitute our expressions for $ \sin(t) $ and $ \cos(t) $ into this identity. By squaring both isolated trigonometric functions, we have:

$ \cos^2(t) = x^2 $ and $ \sin^2(t) = \left(\frac{y}{3}\right)^2 $.

Substituting these into the Pythagorean identity gives us:

$ x^2 + \left(\frac{y}{3}\right)^2 = 1 $.

This equation represents an ellipse, a familiar conic section, demonstrating how the parameter $ t $ can be effectively eliminated through the use of trigonometric identities. The process highlights the importance of recognizing relationships between trigonometric functions and their identities, allowing for a clear transition from parametric to Cartesian forms.

Problem

First eliminate the parameter, then graph the plane curve of the parametric equations.
$x\left(t\right)=2+\cos t$ , $y (t) = - 1 + \sin t$ ; $0\le t\le2\pi$

$\left(x-2\right)^2+\left(y+1\right)^2=1$

Graph showing a circle centered at (2, -1) in the Cartesian plane.

$\left(x-2\right)^2+\left(y+1\right)^2=1$

Graph showing a circle centered at (2, -1) in the Cartesian plane.

$\left(x+2\right)^2+\left(y-1\right)^2=1$

Graph showing a circle centered at (2, -1) in the Cartesian plane.

$\left(x+2\right)^2+\left(y-1\right)^2=1$

Graph showing a circle centered at (2, -1) in the Cartesian plane.

example

Eliminate Parameter: Equations with Trig Example 3

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Eliminate Parameter: Equations with Trig Example 3 Video Summary

In this example, we explore the process of eliminating parameters from parametric equations to derive a rectangular equation. The given parametric equations are x(t) = 3 cos(t) and y(t) = 2 sin(t). To convert these into a rectangular form, we utilize the Pythagorean identity, which states that cos²(t) + sin²(t) = 1.

First, we isolate cos(t) and sin(t) from the parametric equations. From x = 3 cos(t), we can express cos(t) as cos(t) = x/3. Squaring both sides gives us cos²(t) = (x/3)². Similarly, from y = 2 sin(t), we find sin(t) = y/2, leading to sin²(t) = (y/2)².

Substituting these expressions into the Pythagorean identity results in the equation:

\[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \]

This equation represents an ellipse centered at the origin (0, 0). The denominators indicate the lengths of the semi-major and semi-minor axes, with the semi-major axis along the x-axis measuring 3 units and the semi-minor axis along the y-axis measuring 2 units. Thus, the ellipse stretches from -3 to 3 on the x-axis and from -2 to 2 on the y-axis.

However, it is crucial to consider the parameter restrictions. In this case, the parameter t is limited to the interval from π to 2π. This restriction means that we only graph the bottom half of the ellipse, as the top half corresponds to t values from 0 to π. Therefore, the resulting graph will depict only the lower portion of the ellipse, emphasizing the importance of parameter limits in determining the final shape represented by the equations.

concept

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 and ensures that the resulting parametric equations describe the same relationship between $ x $ and $ y $.

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 rectangular 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 $.

In cases where the equation is more complex, such as $ y = (x + 2)^2 - 3 $, a different approach may be necessary. Here, one might choose $ t = x + 2 $. Solving for $ x(t) $ gives $ x(t) = t - 2 $. Substituting this into the equation results in $ y(t) = (t - 2)^2 - 3 $, which can be simplified to $ y(t) = t^2 - 3 $.

It is crucial to avoid domain restrictions when selecting $ t $. Choosing expressions that involve odd powers of $ x $ or simple linear transformations helps prevent issues with negative values or undefined expressions, particularly when dealing with even powers or roots. For example, using $ t = x^2 $ could lead to complications since $ t $ would not be defined for negative values of $ x $.

To verify the correctness of the parametric equations, one can eliminate the parameter $ t $ by substituting back to check if the original rectangular equation is recovered. This serves as a reliable method to ensure that the transformation from rectangular to parametric form has been executed correctly.

example

Parameterizing Equations Example 4

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Parameterizing Equations Example 4 Video Summary

In this discussion, we explore how to parameterize the equation $ y = 2x^3 $ by creating two sets of parametric equations. The first step in parameterization is to define a variable $ t $. A common approach is to set $ t = x $, which simplifies the process. By substituting $ t $ for $ x $, we find that $ x(t) = t $. To derive the corresponding $ y $ equation, we substitute $ x(t) $ back into the original equation, resulting in $ y(t) = 2(t)^3 $. Thus, the first set of parametric equations is:

$ x(t) = t $

$ y(t) = 2t^3 $

For the second set of parametric equations, we can choose a different definition for $ t $. Instead of using $ t = x $, we can set $ t = x^3 $. This choice avoids complications with even powers of $ x $ and maintains a valid domain for both positive and negative values. Solving for $ x $ in terms of $ t $, we find $ x(t) = \sqrt[3]{t} $. Substituting this back into the original equation gives us:

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

which simplifies to $ y(t) = 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 original rectangular equation $ y = 2x^3 $, demonstrating the flexibility in choosing the parameter $ t $ and the various valid approaches to parameterization.

concept

Parameterizing Equations of Circles & Ellipses

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Parameterizing 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 for any angle $t$, the relationship $\cos^2(t) + \sin^2(t) = 1$ holds true. When given an equation like $x^2 + y^2 = 1$, it can be parameterized by setting $x$ and $y$ in terms of trigonometric functions.

For example, if you encounter an equation such as $\frac{x^2}{4} + y^2 = 1$, you can rewrite it in the form of $f(x)^2 + g(y)^2 = 1$. Here, $f(x) = \frac{x}{2}$ and $g(y) = y$. To parameterize this, you can set $\frac{x}{2} = \cos(t)$ and $y = \sin(t)$. Solving for $x$ gives $x = 2\cos(t)$ and for $y$ gives $y = \sin(t)$. Thus, the parametric equations are $x(t) = 2\cos(t)$ and $y(t) = \sin(t)$.

When working with more complex equations, such as $9x^2 + y^2 = 9$, the first step is to manipulate the equation into the standard form. Dividing the entire equation by 9 yields $\frac{x^2}{1} + \frac{y^2}{9} = 1$. This indicates an ellipse, where $f(x) = x$ and $g(y) = \frac{y}{3}$. Setting $x = \cos(t)$ and $\frac{y}{3} = \sin(t)$ leads to $y = 3\sin(t)$. Therefore, the parametric equations for this ellipse are $x(t) = \cos(t)$ and $y(t) = 3\sin(t)$.

In summary, the process of converting Cartesian equations to parametric forms involves identifying the structure of the equation, applying the Pythagorean identity, and solving for $x$ and $y$ in terms of a parameter $t$. This method not only simplifies the equations but also provides a clear geometric interpretation of the relationships between the variables.

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:

Parametric equations express two variables, x and y, in terms of a third variable, typically denoted as t, called the parameter. These equations are written as x(t) and y(t). To graph parametric equations, follow these steps:

Create a table of values with three columns: t, x(t), and y(t).
Choose or use given values for t.
Substitute each t value into the x(t) and y(t) equations to calculate corresponding x and y values.
Plot the resulting (x, y) points on a coordinate plane.
Connect the points in the order of increasing t values, adding arrows to indicate direction.

This process helps visualize the curve or line described by the parametric equations.

To eliminate the parameter t in parametric equations, follow these steps:

Solve one of the parametric equations (either x(t) or y(t)) for t.
Substitute the expression for t into the other equation.
Simplify the resulting equation to express y in terms of x (or vice versa).

For example, if x(t) = t + 1 and y(t) = t² - 3, solve x(t) for t: $t = x - 1$ . Substitute this into y(t): $y = (x - 1)^{2} - 3$ . The resulting equation, $y = x^{2} - 2 x - 2$ , is the rectangular form.

Pythagorean identities are essential for eliminating the parameter in parametric equations involving trigonometric functions. For example, if x(t) = cos(t) and y(t) = sin(t), the identity $\cos^{2} (t) + \sin^{2} (t) = 1$ is used. By squaring both equations and adding them, you eliminate t:

x^{2} + y^{2} = 1

. This results in the familiar equation of a circle. Similarly, for scaled or shifted trigonometric parametric equations, the identity helps derive rectangular equations for ellipses or other conic sections.

To parameterize a rectangular equation, follow these steps:

Choose an expression for t, often setting t = x or t as a simple function of x or y.
Express x and y in terms of t. For example, if y = 2x + 3, set t = x. Then, x(t) = t and y(t) = 2t + 3.
Ensure the parameterization avoids domain restrictions. For instance, avoid square roots or even powers of x that could restrict t to non-negative values.

For more complex equations, such as y = (x + 2)² - 3, you might set t = x + 2, leading to x(t) = t - 2 and y(t) = t² - 3. This process is the reverse of eliminating the parameter.

Parametric equations are widely used in various fields due to their ability to describe motion and curves. Common applications include:

Physics: Modeling projectile motion, where x(t) and y(t) describe horizontal and vertical positions over time.
Engineering: Designing paths for robotic arms or vehicles, where parametric equations define precise trajectories.
Computer Graphics: Creating curves and animations, such as Bézier curves for smooth transitions.
Mathematics: Representing complex curves like ellipses, circles, and parabolas.
Geography: Mapping paths on Earth's surface, such as great-circle routes.

These equations are powerful tools for describing dynamic systems and geometric shapes in a flexible and intuitive way.

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Parametric Equations: Videos & Practice Problems

Introduction to Parametric Equations

Introduction to Parametric Equations Video Summary

Graph the plane curve formed by the parametric equations and indicate its orientation.x(t)=−t+1x\left(t\right)=-t+1; y(t)=t2y\left(t\right)=t^2−2≤t≤2-2\le t\le2

Graph the plane curve formed by the parametric equations and indicate its orientation.x(t)=2t−1x(t)=2t-1; y(t)=2ty(t)=2\sqrt{t}t≥0t\ge0

Eliminating the Parameter

Eliminating the Parameter Video Summary

Eliminate the parameter to rewrite the following as a rectangular equation.x(t)=2t−1x\left(t\right)=2t-1x(t)=2t−1y(t)=t5−2y\left(t\right)=t^5-2y(t)=t5−2

Eliminating the Parameter Example 1

Eliminating the Parameter Example 1 Video Summary

Eliminating the Parameter Example 2

Eliminating the Parameter Example 2 Video Summary

Eliminate Parameter: Equations with Trig

Eliminate Parameter: Equations with Trig Video Summary

First eliminate the parameter, then graph the plane curve of the parametric equations.x(t)=2+costx\left(t\right)=2+\cos tx(t)=2+cost, y(t)=−1+sinty\left(t)=-1+\sin t\right.; 0≤t≤2π0\le t\le2\pi0≤t≤2π

Eliminate Parameter: Equations with Trig Example 3

Eliminate Parameter: Equations with Trig Example 3 Video Summary

Parameterizing Equations

Parameterizing Equations Video Summary

Parameterizing Equations Example 4

Parameterizing Equations Example 4 Video Summary

Parameterizing Equations of Circles & Ellipses

Parameterizing Equations of Circles & Ellipses Video Summary

Write parametric equations for the rectangular equation below.x2+y2=25x^2+y^2=25x2+y2=25

Do you want more practice?

Here’s what students ask on this topic:

What are parametric equations and how do you graph them?

What are parametric equations and how do you graph them?

How do you eliminate the parameter in parametric equations?

How do you eliminate the parameter in parametric equations?

What is the role of Pythagorean identities in parametric equations with trigonometric functions?

What is the role of Pythagorean identities in parametric equations with trigonometric functions?

How do you parameterize a rectangular equation?

How do you parameterize a rectangular equation?

What are some common applications of parametric equations?

What are some common applications of parametric equations?

Your Calculus tutors

Graph the plane curve formed by the parametric equations and indicate its orientation.
$x (t) = - t + 1$ ; $y (t) = t^{2}$
$negative 2 is less than or equal to t is less than or equal to 2$

Graph the plane curve formed by the parametric equations and indicate its orientation.
$x (t) = 2 t - 1$ ; $y (t) = 2 \sqrt{t}$
$t is greater than or equal to 0$

Eliminate the parameter to rewrite the following as a rectangular equation.
$x\left(t\right)=2t-1$
$y\left(t\right)=t^5-2$

First eliminate the parameter, then graph the plane curve of the parametric equations.
$x\left(t\right)=2+\cos t$ , $y (t) = - 1 + \sin t$ ; $0\le t\le2\pi$

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