Question

The equation of a curve is y = x
2 − 6x + k, where k is a constant.
(i) Find the set of values of k for which the whole of the curve lies above the x-axis

Answer 1

Given: a curve y = x² - 6x + k, where k is a constant

To find: the set of values of k for which the whole of the curve lies above the x-axis

Solution:

• The equation of the x-axis is y = 0

• When a curve lies above the x-axis, we must have y > 0
• i.e. x² - 6x + k > 0
• or, (x² - 6x + 9) + (k - 9) > 0
• or, (x - 3)² + (k - 9) > 0

• Since x - 3 ≥ 0 for x ≥ 0, we must have k - 9 > 0, i.e. k > 9 for a positive value of the left hand side expression.

Answer: values of k are > 9.