4.1.1 Determine the x-coordinate of A - NSC Technical Mathematics - Question 4 - 2023 - Paper 1

To show that k = 1, we substitute (k; -3) into the equation of g:

g(k) = -k - 2 = -3$$ Rearranging gives:

k + 2 = 3$$

k = 1$$ Hence, k = 1.

Since k = 1 and B is on the function f, we substitute k into the expression for x:

The x-coordinate of B is thus:

$ext{B} = (1; f(1))$

To show that f(x) = -x² + 2x + 8, we can analyze the structure of the quadratic function. It is evident when given B and A that:

$$A = (-2; f(-2))$$

Using the x-intercepts and properties of parabolas, we can deduce:

$$imes f(x) = a(x + 2)(x - 4)$$

With a = -1, we can verify:

$$f(x) = -1(x + 2)(x - 4) = -x^{2} + 2x + 8$$

To determine the range of f(x), we note that it is a downward-opening parabola with a maximum point at x = 1. Evaluating f(1):

$$f(1) = -1² + 2(1) + 8 = 9$$

Thus, the range is:

$$ext{Range: } y ext{ such that } y ext{ ≤ } 9$$

To solve for f(x) ≥ g(x), we set up the inequality:

$$-x^{2} + 2x + 8 ≥ -x - 2$$

This can be rearranged and solved to find the values of x satisfying the condition.