# Finding the Particle's Path: Graphing a Cartesian Equation

• Fatima Hasan
In summary: Can you tell me how to graph it ?Sure! First, we can rewrite the equation as ##y^2 - x^2 = 4##. This is the equation of a hyperbola with center at the origin and vertices at (0, ±2). The asymptotes of the hyperbola are given by the equations ##y = x## and ##y = -x##. To graph it, you can plot the vertices and asymptotes, and then draw in the hyperbola connecting them.
Fatima Hasan

## Homework Statement

Identify the particle's path by finding a Cartesian equation for it . Graph the Cartesian equation . indicate the portion of the graph traced by the particle .
##x=2sinh t## , ##y=2cosh t## , ##-\infty < t < \infty##

## Homework Equations

##cosh^2 t - sinh^2 t = 1##

## The Attempt at a Solution

##sinh t = \frac{x}{2}##
Square both sides :
##sinh^2 t = \frac{x^2}{4}## (1)
##cosh t = \frac{y}{2}##
Square both sides :
##cosh^2 t = \frac{y^2}{4}## (2)
(2) - (1) :
##cosh^2 t - sinh^2 t = 1##
##\frac{y^2}{4} - \frac{x^2}{4} = 1##
put x= 0 → y=±2

Is my answer correct ?

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Fatima Hasan said:
Is my answer correct ?

What do you know about the possible values of ##\cosh t## for ##-\infty < t < \infty##?

George Jones said:
What do you know about the possible values of ##\cosh t## for ##-\infty < t < \infty##?
The Cartesian equation forms a parbola opening up . Right ?

I just noticed something else.
$$\frac{x^2}{4} - \frac{y^2}{4} =1$$
is not correct, but I think this is a "typo".

Fatima Hasan said:
The Cartesian equation forms a parbola opening up . Right ?
No. The graph you show is not a parabola.

Fatima Hasan said:
Identify the particle's path by finding a Cartesian equation for it .
Graph the Cartesian equation . indicate the portion of the graph traced by the particle .
##x=2sinh t## , ##y=2cosh2## , ##-\infty < t < \infty##
Should the second equation be ##y = 2\cosh(t)##?

Mark44 said:
Should the second equation be ##y = 2\cosh(t)##?
yeah

I agree with @George Jones's comment in post #4.
Fatima Hasan said:
(2) - (1) :
##cosh^2 t - sinh^2 t = 1##
##\frac{x^2}{4} - \frac{y^2}{4} = 1##
Take a closer look at the work I've quoted above.

##\frac{y^2}{4} - \frac{x^2}{4}## forms a hyperbola and to graph it , we should go 2 units up from center point .
##cosh ( t) ## for ##-\infty < t < \infty## is always positive .

Mark44 said:
I agree with @George Jones's comment in post #4.
Take a closer look at the work I've quoted above.
##\frac{y^2}{4} - \frac{x^2}{4} = 1##

Fatima Hasan said:
##\frac{y^2}{4} - \frac{x^2}{4} = 1##
Yes, that looks better.

Fatima Hasan said:
##\frac{y^2}{4} - \frac{x^2}{4} = 1## forms a hyperbola

Yes, a big hint is given by the names of the functions in the original equations.

Fatima Hasan
Mark44 said:
Yes, that looks better.

Is it correct ?

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Fatima Hasan said:
View attachment 231720
Is it correct ?
No.
The equation ##\frac{y^2}{4} - \frac{x^2}{4} = 1## is NOT a parabola. The graph you showed appears to be the graph of ##y = x^2 + 2##, which is completely unrelated to your equation.

Mark44 said:
The equation ##\frac{y^2}{4} - \frac{x^2}{4} = 1## is NOT a parabola.

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Fatima Hasan said:
Bingo!
(Meaning, yes, that's it.)

Fatima Hasan
Mark44 said:
Bingo!
(Meaning, yes, that's it.)
Thanks for your help.

## 1. What is a Cartesian equation?

A Cartesian equation is a mathematical representation of a relationship between two variables using the Cartesian coordinate system. It is named after the French mathematician René Descartes who introduced the system.

## 2. How do you graph a Cartesian equation?

To graph a Cartesian equation, you need to plot the points that satisfy the equation on a coordinate plane. This can be done by assigning values to the variables and plotting the corresponding points. Then, connect the points to create a visual representation of the equation.

## 3. What is the difference between a Cartesian equation and a polar equation?

A Cartesian equation uses the x and y coordinates to represent a relationship between two variables, while a polar equation uses the distance from the origin and the angle from a fixed reference line. In other words, Cartesian equations use rectangular coordinates while polar equations use polar coordinates.

## 4. What are the applications of graphing Cartesian equations?

Graphing Cartesian equations is useful in various fields such as physics, engineering, economics, and statistics. It can help visualize and analyze relationships between variables and make predictions based on the graph.

## 5. Can you graph a Cartesian equation with more than two variables?

No, a Cartesian equation can only represent a relationship between two variables. If there are more than two variables, the equation cannot be graphed on a two-dimensional coordinate plane. However, it is possible to graph an equation with three variables using a three-dimensional coordinate system.

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