The graphs of the functions $$f(x) = - (x + 3)^2 + 4$$ and $$g(x) = x + 5$$ are drawn below - NSC Mathematics - Question 5 - 2023 - Paper 1

The range of the function $f$ is $y \leq 4$ or $y \in (-\infty, 4]$.

To find the x-coordinates of A and B, set $f(x) = g(x)$:

$-(x + 3)^2 + 4 = x + 5$

Rearranging gives:

$-(x + 3)^2 - x - 1 = 0$

This can be simplified to:

$-(x^2 + 6x + 9) - x - 1 = 0$

Thus, we have:

$-x^2 - 7x - 10 = 0$

Factoring, we find:

$x^2 + 7x + 10 = 0$

The factors are $(x + 5)(x + 2) = 0$, leading to:

$x = -5, -2$.

Rearranging the equation gives:

$-x - c + 4 = x + 5$

This simplifies to:

$-2x - c + 4 - 5 = 0$

Or,

$-2x - c - 1 = 0$,

which leads to:

$x = \frac{-c - 1}{-2}$.

To have one negative and one positive root, the graph must cross the x-axis in such a way that the associated values of c must satisfy the inequality: $-5 < c < -2$.

The distance between the two functions is given by:

$D(x) = f(x) - g(x)$,

Substituting the respective functions gives:

$D(x) = - (x + 3)^2 + 4 - (x + 5)$,

This simplifies to:

$D(x) = - (x + 3)^2 - x - 1$.

To calculate the maximum distance, we can consider the first derivative and find critical points. The maximum occurs at:

$\text{Max: } -2x - 7 = 0$

This results in:

$x = -\frac{7}{2}$.