Consider the curve parameterized by 𝑥=𝑡^2−9,𝑦=𝑡^2−8𝑡. Determine the values of that correspond to the portion of the curve in each of the four quadrants.

Question

Determine the values of  that correspond to the portion of the curve in each of the four quadrants.

Quadrant I: t∈

Quadrant II: t∈

Quadrant III: 𝑡∈

Quadrant IV: 𝑡∈

Answer 1

If x = t² - 9 = (t - 3)(t-3), x > 0 for (-∞, -3) ∪ (3, ∞) and x < 0 for (-3, 3)

If y = (t² - 8t) = t(t - 8), y > 0 for (-∞, 0) ∪ (8, ∞) and y < 0 for (0, 8)

Thus, consider the intervals (-∞, -3), (-3, 0), (0, 3), (3, 8), and (8, ∞)

On (-∞, -3) ∪ (8, ∞), x > 0 and y > 0, so we are in Quadrant I

On (-3, 0), x < 0 and y > 0, so we are in Quadrant II

On (0, 3), x< 0 and y < 0, so we are in Quadrant III

On (3, 8), x > 0 and y < 0, so we are in Quadrant IV.