A class is trying to eliminate t from the parametric equations x equals t superscript 2 baseline plus 3 and y = 4t. Beth says that she can write t equals start root x minus 3 end root to eliminate the parameter. Why is this wrong?

Question

(1 point)
Responses

She should have added 3 to x, not subtracted.
She should have added 3 to x , not subtracted.

She should always solve for t as a function of y.
She should always solve for t as a function of y .

She should have taken both the positive and negative square root.
She should have taken both the positive and negative square root.

She should first substitute y for t before solving.

Answer 1

Beth's solution is incorrect because:

She should have taken both the positive and negative square root.

The equation \( t = \sqrt{x - 3} \) only considers the positive square root, which does not account for negative values of \( t \) that can produce the same \( x \) value. Since the original parametric equations do not restrict \( t \) to positive values, it is necessary to consider both the positive and negative roots when expressing \( t \) in terms of \( x \). This means that the correct expression should actually be \( t = \pm \sqrt{x - 3} \).