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 function $f(x) = -(x + 3)^2 + 4$ is a downward-opening parabola. The maximum value occurs at the turning point, which is $4$. As $x$ approaches infinity, $f(x)$ tends towards negative infinity.

Thus, the range of $f$ is:

$(- ext{∞}, 4]$.

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

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

Rearranging gives:

$-(x^2 + 6x + 9) + 4 = x + 5$

$-x^2 - 6x - 9 + 4 = x + 5$

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

Multiplying through by -1:

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

Factoring the quadratic:

$(x + 5)(x + 2) = 0$

Setting each factor to zero gives:

ightarrow x = -5$$ $$x + 2 = 0 ightarrow x = -2$$ Thus, the x-coordinates of A and B are $-5$ and $-2$, respectively.

For the equation $-(x + 3)^2 + 4 + c = 0$ to have one negative and one positive root, the discriminant must be positive and the vertex of the parabola should be above the x-axis:

1. The vertex occurs at $x = -3$ leading to a $y$-value: $f(-3) = -(0)^2 + 4 + c = 4 + c$

To have one root at $x = 0$ (crossing the axis), we set:

ightarrow c = -4$$ 2. The parabola opens downwards; thus, for one positive and one negative root, we need: $$c < -4$$ Thus, the inequality for $c$ is: $$c < -4$$.

First, express the distance function:

$d(x) = |f(x) - g(x)| = |[-(x + 3)^2 + 4] - (x + 5)|$

This simplifies to:

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

To find the maximum distance, calculate the critical points and evaluate on the interval. Once the maximum value is determined as $k$, substitute into $h(x)$:

$h(x) = g(x) + k = (x + 5) + k$.

Now set it out as:

$H(x) = x + (5 + k)$.