Question

CALC HELP!!! WILL MARK BRAINLIEST
Consider the parametric equations below.


x = t2 − 2, y = t + 1, −3 ≤ t ≤ 3


(a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases.


(b) Eliminate the parameter to find a Cartesian equation of the curve. for −2 ≤ y ≤ 4

Answer 1

Answer:

(a) In attachment

(b) x = y² - 2y - 1,    −2 ≤ y ≤ 4

Step-by-step explanation:

(a)

The graph of the given parametric equation is given in the attachment.

The direction in which the curve is traced as t increases, is indicated by black arrows.

(b)

To eleminate the parameter t, we simultaneously solve both the equations.

So, we have the equations:

x = t² - 2   ----- equation (1)

y = t + 1   ----- equation (2)

So, from equation (2), we have:

t = y - 1

Substituting this in equation (1), we get:

x = (y - 1)² - 2

x = y² - 2y + 1 - 2

x = y² - 2y - 1

Now, for limits of y, we use equation (2)

For initial limit, t = -3

y = - 3 + 1 = - 2

For final limit, t = 3

y = 3 + 1 = 4

Therefore, the final relation after eliminating t is:

x = y² - 2y - 1,    −2 ≤ y ≤ 4

Answer 2

The Cartesian equation of the curve after eliminating the parameter t is expressed below.

$x=y^2-2y-1$

For, the y values of, $-2 \leq y \leq 4$

What is a parametric equation?

The parametric equation is the type of equation  in which the variable which is in depended on on is known as parameter. The dependent function in this equation is defined as the continuous function of that variable.

The first equation given in the problem is,

$x = t^2 - 2 \\t^2=x+2$                    

The second equation given in the problem is,

$y = t + 1\\t=y-1$

Put the values of t in the modified first equation as,

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

The values of t are between -3 to 3.

$-3 \leq t \leq 3$

The value of t is -3 then the value of y will be -2 and when the value of t is 4 then the value of y will be 4 from equation 2.

Hence, the Cartesian equation of the curve after eliminating the parameter t is expressed below.

$x=y^2-2y-1$

For, the y values of, $-2 \leq y \leq 4$

Learn more about the parametric equation here;

https://brainly.com/question/21845570