Express the given parametric curve by an equation in x and y, then sketch the curve.
(a) x = 2 tan t, y = 3 sec2 t − 4, for −
�
2
< t <
�
2
.
(b) x = 2et + 1, y = 3et + 2.
To express the given parametric curves as equations in x and y, we need to eliminate the parameter t.
(a) x = 2tan(t), y = 3sec^2(t) - 4
To eliminate t, we can use the relationship between sec^2(t) and tan(t):
sec^2(t) = 1 + tan^2(t)
Substitute sec^2(t) with (1 + tan^2(t)) in the equation for y:
y = 3(1 + tan^2(t)) - 4
= 3tan^2(t) + 3 - 4
= 3tan^2(t) - 1
Now we have the equation in terms of x and y:
x = 2tan(t)
y = 3tan^2(t) - 1
To sketch the curve, we can plot points for different values of t and connect them to visualize the curve. Choose a range for t, such as -π/2 to π/2, and calculate corresponding values of x and y using the equations above. Then plot those points on a coordinate plane and connect them to form the curve.
(b) x = 2e^t + 1, y = 3e^t + 2
In this case, there is no parameter to eliminate. The given equations directly represent the curve in terms of x and y.
To sketch the curve, again choose a range for t, such as -1 to 1, and calculate corresponding values of x and y using the equations above. Plot those points on a coordinate plane and connect them to form the curve.