15–30. Working with parametric equations Consider the followin...

Question

15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.

x = 8 + 2t, y = 1; −∞ < t < ∞

Accepted Answer

Welcome back everyone. Given the parametric equations X equals 3 minus 4 C and Y equals 5, for C between negative infinity and infinity, eliminate the parameter to find an equation relating X and Y. Then describe the curve represented by this equation and specify the positive orientation. So, for this problem, let's go ahead and show that Y is equal to 5. And what we have to notice is that there is no parameter in this equation, right? So, in reality, because we're looking for the equation Y of X, and there is no parameter T in this equation, we can essentially conclude that we already have the The equation that we want to get in this problem, it is Y equals 5, so we're going to label it as our final answer for the first part of the problem. Now we're going to describe this curve. Let's recall that Y equals A represents a horizontal line. Which intersects the y axis. At y equals a. So what we're going to do from here is simply state that it is a horizontal. Line With Why? Intercept We're going to specify the coordinates. Let's recall that Y intercepts have X of 0, and in this case, our intercept has Y coordinate of 5. So the Y intercept has a form of 0.5, and we also want to state that it is traced. In some direction as T increases, right? So T is between negative infinity and infinity. We want to focus on the X coordinated self. We know that X is equal to 3 minus 4 T. So if T approaches negative infinity, then X is going to approach 3 minus 4 multiplied by negative infinity. So 3 plus infinity gives us positive infinity, right? And when T approaches positive infinity, X is going to approach 3 minus 4 multiplied by infinity. So that's 3 minus infinity or basically negative infinity. Notice that as T increases, X goes from positive infinity to negative infinity, meaning we're tracing our line from right to left, right? Because if we visualize the x-axis, we have positive infinity on the right, negative infinity on the left, and if we're moving from positive infinity to negative infinity, it basically means from right to left. So we're tracing our horizontal line from right to left as T increases. We can say traced from right. To Left As T increases. That's our final answer and thank you for watching.

15–30. Working with parametric equations Consider the following parametric equations.
a. Eliminate the parameter to obtain an equation in x and y.
b. Describe the curve and indicate the positive orientation.

x = 8 + 2t, y = 1; −∞ < t < ∞

Key Concepts

Parametric Equations

Eliminating the Parameter

Curve Orientation

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