1.) Consider the parametric equations below.
x = t2 − 2, y = t + 1, −3 ≤ t ≤ 3
(b) Eliminate the parameter to find a Cartesian equation of the curve.
for −2 ≤ y ≤ 4.
2.) Consider the following.
x = et − 8, y = e2t
(a) Eliminate the parameter to find a Cartesian equation of the curve.
3.) Consider the parametric equations below.
x = 1 + t, y = 5 − 4t, −2 ≤ t ≤ 3
(b) Eliminate the parameter to find a Cartesian equation of the curve.
for -1 ≤ x ≤ 4

The Cartesian equation of the curve is y = -4x + 9 for -1 ≤ x ≤ 4.

To eliminate the parameter t, we can isolate t in the equation x = t^2 - 2 and substitute it into the equation for y. This gives us:

y = t + 1

t = x + 2

y = x + 3

So the Cartesian equation of the curve is y = x + 3 for -2 ≤ y ≤ 4.

We can eliminate t by taking the natural logarithm of both x and y. This gives us:

ln x = t - 8

ln y = 2t

Solving for t in the first equation and substituting it into the second equation, we get:

t = ln(x) + 8

y = e^(2ln(x)+16) = x^2e^16

So the Cartesian equation of the curve is y = x^2e^16.

To eliminate t, we can again isolate it in the equation for x and substitute it into the equation for y. This gives us:

x = t + 1

t = x - 1

y = 5 - 4(x - 1) = -4x + 9

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The Cartesian equation of the curve is y = 1 - 4x, for -1 ≤ x ≤ 4.

To eliminate the parameter t, we can solve for t in terms of x and substitute it into the equation for y:

t = ±√(x + 2)

y = t + 1 = ±√(x + 2) + 1

Squaring both sides and simplifying, we get:

(x + 2) = (y - 1)²

So the Cartesian equation of the curve is:

x = (y - 1)² - 2, for -2 ≤ y ≤ 4.

To eliminate the parameter t, we can take the natural logarithm of both sides of the equation for y:

ln y = 2 ln e + t ln e = 2 ln e + ln x

ln y = ln (xe²)

y = xe²

So the Cartesian equation of the curve is:

y = x^2e^2.

To eliminate the parameter t, we can solve for t in terms of x and substitute it into the equation for y:

t = 1 + x

y = 5 - 4t = 1 - 4x

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