Question

Describe the similarities and differences between the parametric equations \(x=t, y=t^{2}\) and \(x=-t, y=t^{2},\) where \(t \geq 0\) in each case.

Short answer

Answer: The similarities between the parametric equations are that they both have the same \(y\)-component expression \(y=t^{2}\), resulting in the same \(y\)-values for a given value of \(t\). The differences lie in their \(x\)-components, with the first equation producing points in the first and second quadrants (right side of the coordinate plane), and the second equation producing points in the third and fourth quadrants (left side of the coordinate plane).

Step-by-step solution

Identify the parametric equations

Given the two parametric equations: 1. \(x = t, y = t^2\) where \(t \geq 0\) 2. \(x = -t, y = t^2\) where \(t \geq 0\)

Analyze similarities

We can see that in both parametric equations, the \(y\)-component has the same expression, \(y = t^{2}\). It means that for both equations, the \(y\)-value of a point will be the same for a given value of \(t\).

Analyze differences

While the \(y\)-components are the same, the \(x\)-components are different. Let's consider their differences: 1. In the first parametric equation, \(x = t\). As \(t\) is non-negative, the \(x\)-values will also be non-negative. 2. In the second parametric equation, \(x = -t\). Since \(t\) is non-negative, the \(x\)-values will be non-positive. Thus, in the first equation, the points will lie in the first and second quadrants (right side of the coordinate plane), while in the second equation, the points will lie in the third and fourth quadrants (left side of the coordinate plane).

Conclusion

The similarities between the parametric equations \(x=t, y=t^{2}\) and \(x=-t, y=t^{2}\), where \(t \geq 0\) in each case, are that they both have the same \(y\)-component expression \(y=t^{2}\). This means that for a given value of \(t\), both equations will generate points with the same \(y\)-values. The differences between the two parametric equations are that while the first equation will produce points in the first and second quadrants (right side of the coordinate plane), the second equation will produce points in the third and fourth quadrants (left side of the coordinate plane). This difference is due to the opposite signs in their \(x\)-components: \(x=t\) and \(x=-t\).