Question

Which of the following provides a parametrization of the graph of y = 2x³ - x - 4:

(i) (2t, 16t³ - 2t - 4)
(ii) (t+1, 2t³ + 6t² + 5t - 3)

Answer 1

The correct parametrization of the graph of
`\(y = 2x³ - x - 4\) is \((t+1, 2t³ + 6t² + 5t - 3)\).`

Step-by-step explanation:

The given parametrization
`\((t+1, 2t³ + 6t² + 5t - 3)\)` can be used to express
`\(x\) and \(y\). In this case, \(x = t + 1\) and \(y = 2t³ + 6t² + 5t - 3\).`By substituting
`\(t + 1\) for \(x\) in the original function \(y = 2x³ - x - 4\),` we can verify its correctness.

`Now, substitute \(t + 1\) for \(x\) in \(y = 2x³ - x - 4\):`

`\[y = 2(t + 1)³ - (t + 1) - 4\]`

Expand and simplify:

`\[y = 2(t³ + 3t² + 3t + 1) - t - 1 - 4\]`

`\[y = 2t³ + 6t² + 5t - 3\]`

Thus, the expression matches the original function
`\(y = 2x³ - x - 4\).`This confirms that the parametrization
`\((t+1, 2t³ + 6t² + 5t - 3)\)`correctly represents the graph of
`\(y = 2x³ - x - 4\).`