sketch the curve represented by the parametric equations. Then eliminate and write the corresponding rectangular equation whose graph represents the curve.
10. x=t2-6
y=1/2t-1
well, t = 2y+2, so use that
sketch the curve represented by the parametric equations. Then eliminate and write the corresponding rectangular equation whose graph represents the curve.
10. x=t2-6
y=1/2t-1
how do I use
t = 2y+2?
come on. just substitute it into the other equation.
t = 2y+2
x = t^2-6 = (2y+2)^2-6 = ...
It is clearly a horizontal parabola
To sketch the curve represented by the parametric equations and eliminate and write the corresponding rectangular equation, follow these steps:
Step 1: Sketch the curve
a. Plot points on the curve for different values of t.
b. To do this, choose a range of values for t, such as -5 to 5, and calculate the corresponding x and y values using the given parametric equations.
c. Once you have several points, plot them on a graph.
d. Connect the plotted points smoothly to get an idea of the shape of the curve.
Step 2: Eliminate the parameter t
a. To eliminate the parameter t, rearrange one of the equations to solve for t in terms of x (or y).
b. Let's rearrange the first equation, x = t^2 - 6, to solve for t:
x + 6 = t^2
√(x + 6) = t
c. Substitute this value of t into the second equation, y = 1/2t - 1:
y = 1/2(√(x + 6)) - 1
d. Simplify the equation to obtain the rectangular equation, which represents the curve.
Therefore, the corresponding rectangular equation for the given parametric equations x = t^2 - 6 and y = 1/2t - 1 is y = 1/2(√(x + 6)) - 1. This equation represents the curve.