8.7 Parametric equations: graphs

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In this section you will:

Graph plane curves described by parametric equations by plotting points.
Graph parametric equations.

It is the bottom of the ninth inning, with two outs and two men on base. The home team is losing by two runs. The batter swings and hits the baseball at 140 feet per second and at an angle of approximately $45°$ to the horizontal. How far will the ball travel? Will it clear the fence for a game-winning home run? The outcome may depend partly on other factors (for example, the wind), but mathematicians can model the path of a projectile and predict approximately how far it will travel using parametric equations . In this section, we’ll discuss parametric equations and some common applications, such as projectile motion problems.

Photo of a baseball batter swinging. — Parametric equations can model the path of a projectile. (credit: Paul Kreher, Flickr)

Graphing parametric equations by plotting points

In lieu of a graphing calculator or a computer graphing program, plotting points to represent the graph of an equation is the standard method. As long as we are careful in calculating the values, point-plotting is highly dependable.

How To

Given a pair of parametric equations, sketch a graph by plotting points.

Construct a table with three columns: $t, x (t), and y (t) .$
Evaluate $x$ and $y$ for values of $t$ over the interval for which the functions are defined.
Plot the resulting pairs $(x, y) .$

Sketching the graph of a pair of parametric equations by plotting points

Sketch the graph of the parametric equations $x (t) = t^{2} + 1, y (t) = 2 + t .$

Construct a table of values for $t, x (t),$ and $y (t),$ as in [link] , and plot the points in a plane.

$t$	$x (t) = t^{2} + 1$	$y (t) = 2 + t$
$- 5$	$26$	$- 3$
$- 4$	$17$	$- 2$
$- 3$	$10$	$- 1$
$- 2$	$5$	$0$
$- 1$	$2$	$1$
$0$	$1$	$2$
$1$	$2$	$3$
$2$	$5$	$4$
$3$	$10$	$5$
$4$	$17$	$6$
$5$	$26$	$7$

The graph is a parabola with vertex at the point $(1, 2),$ opening to the right. See [link] .

Graph of the given parabola opening to the right.

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Sketch the graph of the parametric equations $x = \sqrt{t}, y = 2 t + 3, 0 \leq t \leq 3.$

Graph of the given parametric equations with the restricted domain - it looks like the right half of an upward opening parabola.

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Sketching the graph of trigonometric parametric equations

Construct a table of values for the given parametric equations and sketch the graph:

\begin{array}{l} \begin{array}{l} x = 2 \cos t \\ y = 4 \sin t \end{array} \end{array}

Construct a table like that in [link] using angle measure in radians as inputs for $t,$ and evaluating $x$ and $y .$ Using angles with known sine and cosine values for $t$ makes calculations easier.

$t$	$x = 2 \cos t$	$y = 4 \sin t$
0	$x = 2 \cos (0) = 2$	$y = 4 \sin (0) = 0$
$\frac{π}{6}$	$x = 2 \cos (\frac{π}{6}) = \sqrt{3}$	$y = 4 \sin (\frac{π}{6}) = 2$
$\frac{π}{3}$	$x = 2 \cos (\frac{π}{3}) = 1$	$y = 4 \sin (\frac{π}{3}) = 2 \sqrt{3}$
$\frac{π}{2}$	$x = 2 \cos (\frac{π}{2}) = 0$	$y = 4 \sin (\frac{π}{2}) = 4$
$\frac{2 π}{3}$	$x = 2 \cos (\frac{2 π}{3}) = - 1$	$y = 4 \sin (\frac{2 π}{3}) = 2 \sqrt{3}$
$\frac{5 π}{6}$	$x = 2 \cos (\frac{5 π}{6}) = - \sqrt{3}$	$y = 4 \sin (\frac{5 π}{6}) = 2$
$π$	$x = 2 \cos (π) = - 2$	$y = 4 \sin (π) = 0$
$\frac{7 π}{6}$	$x = 2 \cos (\frac{7 π}{6}) = - \sqrt{3}$	$y = 4 \sin (\frac{7 π}{6}) = - 2$
$\frac{4 π}{3}$	$x = 2 \cos (\frac{4 π}{3}) = - 1$	$y = 4 \sin (\frac{4 π}{3}) = - 2 \sqrt{3}$
$\frac{3 π}{2}$	$x = 2 \cos (\frac{3 π}{2}) = 0$	$y = 4 \sin (\frac{3 π}{2}) = - 4$
$\frac{5 π}{3}$	$x = 2 \cos (\frac{5 π}{3}) = 1$	$y = 4 \sin (\frac{5 π}{3}) = - 2 \sqrt{3}$
$\frac{11 π}{6}$	$x = 2 \cos (\frac{11 π}{6}) = \sqrt{3}$	$y = 4 \sin (\frac{11 π}{6}) = - 2$
$2 π$	$x = 2 \cos (2 π) = 2$	$y = 4 \sin (2 π) = 0$

[link] shows the graph.

Graph of the given equations - a vertical ellipse.

By the symmetry shown in the values of $x$ and $y,$ we see that the parametric equations represent an ellipse . The ellipse is mapped in a counterclockwise direction as shown by the arrows indicating increasing $t$ values.

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Graph the parametric equations: $x = 5 \cos t, y = 3 \sin t .$

Graph of the given equations - a horizontal ellipse.

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Graphing parametric equations and rectangular form together

Graph the parametric equations $x = 5 \cos t$ and $y = 2 \sin t .$ First, construct the graph using data points generated from the parametric form . Then graph the rectangular form of the equation. Compare the two graphs.

Construct a table of values like that in [link] .

$t$	$x = 5 \cos t$	$y = 2 \sin t$
$0$	$x = 5 \cos (0) = 5$	$y = 2 \sin (0) = 0$
$1$	$x = 5 \cos (1) \approx 2.7$	$y = 2 \sin (1) \approx 1.7$
$2$	$x = 5 \cos (2) \approx −2.1$	$y = 2 \sin (2) \approx 1.8$
$3$	$x = 5 \cos (3) \approx −4.95$	$y = 2 \sin (3) \approx 0.28$
$4$	$x = 5 \cos (4) \approx −3.3$	$y = 2 \sin (4) \approx −1.5$
$5$	$x = 5 \cos (5) \approx 1.4$	$y = 2 \sin (5) \approx −1.9$
$−1$	$x = 5 \cos (−1) \approx 2.7$	$y = 2 \sin (−1) \approx −1.7$
$−2$	$x = 5 \cos (−2) \approx −2.1$	$y = 2 \sin (−2) \approx −1.8$
$−3$	$x = 5 \cos (−3) \approx −4.95$	$y = 2 \sin (−3) \approx −0.28$
$−4$	$x = 5 \cos (−4) \approx −3.3$	$y = 2 \sin (−4) \approx 1.5$
$−5$	$x = 5 \cos (−5) \approx 1.4$	$y = 2 \sin (−5) \approx 1.9$

Plot the $(x, y)$ values from the table. See [link] .

Graph of the given ellipse in parametric and rectangular coordinates - it is the same thing in both images.

Next, translate the parametric equations to rectangular form. To do this, we solve for $t$ in either $x (t)$ or $y (t),$ and then substitute the expression for $t$ in the other equation. The result will be a function $y (x)$ if solving for $t$ as a function of $x,$ or $x (y)$ if solving for $t$ as a function of $y .$

\begin{array}{l} x = 5 \cos t \\ \frac{x}{5} = \cos t & Solve for \cos t . \\ y = 2 \sin t \begin{matrix}  \end{matrix} & Solve for \sin t . \\ \frac{y}{2} = \sin t \end{array}

Then, use the Pythagorean Theorem .

\begin{array}{r} \cos^{2} t + \sin^{2} t = 1 \\ {(\frac{x}{5})}^{2} + {(\frac{y}{2})}^{2} = 1 \\ \frac{x^{2}}{25} + \frac{y^{2}}{4} = 1 \end{array}

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Questions & Answers

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In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,

Ezea

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other things being equal

AI-Robot

When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab

Kelo

Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)

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What is different between quantity demand and demand?

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Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded

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Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.

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Jabir

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Asui

it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs

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In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p

Cornelius

Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .

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Answer

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Jabir

the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema

Gsbwnw Reply

suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product

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Source: OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6

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