sketch the curve represented by the parametric equations. Then eliminate and write the corresponding rectangular equation whose graph represents the curve.

x=(√t^2+2)
y=t/4

To sketch the curve represented by the parametric equations, we can plot several points and then connect them smoothly. Let's start by choosing different values for t and calculating the corresponding x and y coordinates:

For t = -4:
x = √((-4)^2 + 2) = √18 ≈ 4.24
y = -4/4 = -1

For t = -3:
x = √((-3)^2 + 2) = √11 ≈ 3.32
y = -3/4 = -0.75

For t = -2:
x = √((-2)^2 + 2) = √6 ≈ 2.45
y = -2/4 = -0.5

For t = -1:
x = √((-1)^2 + 2) = √3 ≈ 1.73
y = -1/4 = -0.25

For t = 0:
x = √((0)^2 + 2) = √2 ≈ 1.41
y = 0/4 = 0

For t = 1:
x = √(1^2 + 2) = √3 ≈ 1.73
y = 1/4 = 0.25

For t = 2:
x = √(2^2 + 2) = √6 ≈ 2.45
y = 2/4 = 0.5

For t = 3:
x = √(3^2 + 2) = √11 ≈ 3.32
y = 3/4 = 0.75

For t = 4:
x = √(4^2 + 2) = √18 ≈ 4.24
y = 4/4 = 1

Now, let's plot these points on a graph and connect them smoothly:

(4.24, -1) - (3.32, -0.75) - (2.45, -0.5) - (1.73, -0.25) - (1.41, 0) - (1.73, 0.25) - (2.45, 0.5) - (3.32, 0.75) - (4.24, 1).

Next, to eliminate the parameter, we can isolate t in the second equation and then substitute it into the first equation:

y = t/4
4y = t
t = 4y

By substituting t = 4y into the first equation:

x = √((4y)^2 + 2)
x = √(16y^2 + 2)

Thus, the corresponding rectangular equation is:

x = √(16y^2 + 2)

To sketch the curve represented by the given parametric equations and eliminate to obtain the corresponding rectangular equation, follow these steps:

Step 1: Understand the Parameter Range
First, determine the range of the parameter, which is usually denoted by "t". In this case, the parameter "t" can take any real value.

Step 2: Plot Points Using the Parameter
To sketch the curve, substitute different values of "t" into the parametric equations to obtain corresponding (x, y) coordinates. It's best to select a few values of "t" that cover a significant portion of the parameter range.

For example, let's choose three values of "t":
1) t = -2
2) t = 0
3) t = 2

Substituting these values into the parametric equations gives us the following coordinates:
1) t = -2: x = (√((-2)^2 + 2)) = √6, y = -2/4 = -0.5
2) t = 0: x = (√(0^2 + 2)) = √2, y = 0/4 = 0
3) t = 2: x = (√((2)^2 + 2)) = √6, y = 2/4 = 0.5

Plotting these three points (-√6, -0.5), (√2, 0), and (√6, 0.5) on the Cartesian plane forms a portion of the curve.

Step 3: Sketch the Curve
Connect the plotted points smoothly with a curve, making sure to extend the curve beyond the plotted points based on the pattern observed.

In this case, since the equation for x is √t^2+2, it represents a curve with a horizontal symmetry with respect to the y-axis. The values of x increase with increasing |t|, meaning the curve moves to the right in the positive x-direction.

Similarly, the equation for y is t/4, which represents a curve that moves vertically upward with increasing |t|.

Based on these observations, sketch a curve that starts at (-√6, -0.5), passes through (√2, 0), and ends at (√6, 0.5). The curve should be smooth, increasing as it moves to the right while also moving upward.

Step 4: Eliminate to Obtain Rectangular Equation
To obtain the rectangular equation, eliminate the parameter "t" from the given parametric equations. In this case, we will eliminate "t" to find the relationship between x and y.

From the given equations, x = (√t^2 + 2) and y = t/4, we can solve for "t" in terms of x or y.

From the equation y = t/4, we can isolate "t" by multiplying both sides by 4:
4y = t

Substitute this value of "t" into the equation for x:
x = (√((4y)^2 + 2))
Simplifying,
x = (√(16y^2 + 2))

This is the rectangular equation representing the curve.

In summary, to sketch the curve represented by the parametric equations, plot points generated by substituting different values of "t" into the equations. Then connect the points smoothly to form the curve. To obtain the corresponding rectangular equation, eliminate the parameter "t" by solving one equation for "t" and substituting into the other equation.