Let's analyze the two pairs of parametric equations: ### First pair: - \(x_1 = t^2\) - \(y_1 = t^6\) ### Second pair: - \(x_2 = t^3\) - \(y_2 = t^9\) You are asking if they **have the same graph** (i.e., describe the same curve in the \(xy\)-plane). --- ## Analyzing the relationship: Let's find the relationship between \(x\) and \(y\) in both cases. ### First pair: \(x_1 = t^2,\, y_1 = t^6\) Express \(y_1\) in terms of \(x_1\): \[ y_1 = t^6 = (t^2)^3 = (x_1)^3 \] So, the curve is **\(y = x^3\)**. --- ### Second pair: \(x_2 = t^3,\, y_2 = t^9\) Express \(y_2\) in terms of \(x_2\): \[ y_2 = t^9 = (t^3)^3 = (x_2)^3 \] Again, the curve is **\(y = x^3\)**. --- ## Are the graphs exactly the same? Let's check the possible subtleties: - **Domain for \(x_1 = t^2\):** \(x_1 \geq 0\) for all real \(t\). - So for the first pair, only the right half (\(x \geq 0\)) of the curve \(y = x^3\) is traced. - **Domain for \(x_2 = t^3\):** \(x_2\) can be any real number (since \(t\) can be any real number). - So the second pair traces the **entire curve** \(y = x^3\). --- ## **Conclusion** **The two parametric equations do NOT have the same graph.** - **First pair (\(x = t^2, y = t^6\))** traces only the part of \(y = x^3\) for \(x \geq 0\). - **Second pair (\(x = t^3, y = t^9\))** traces the entire curve \(y = x^3\), for all real \(x\). **In summary:** > The parametric equations \(x = t^2\), \(y = t^6\) and \(x = t^3\), \(y = t^9\) **do not have the same graph**. The first only describes the right half (\(x \geq 0\)) of the curve \(y = x^3\), while the second describes the entire curve.