Given: Functions $f$ and $h$ defined by $f(x) = -2(-x^3) + 18$ and $h(x) = 2x + c$ 4.1.1 Write down the coordinates of the turning point of $f$ - NSC Technical Mathematics - Question 4 - 2024 - Paper 1

To find the $x$-intercepts of $f$, set $f(x) = 0$:

The sketch should include:

• The turning point at $(0, 18)$.
• The $x$-intercept at (-rac{3}{2}, 0).
• Mark the y-intercept at $(0, 18)$ as well.

Plot the curve illustrating the shape of cubic functions, which should increase through the turning point.

Since the point A $(5, t)$ is the intersection of functions $f$ and $h$, we substitute $x=5$ into both functions:

For $f$:
$f(5) = -2(-5^3) + 18 = -2(-125) + 18 = 250 + 18 = 268$

Thus, $t = 268$.

Using the point of intersection $(5, t)$ on the function $h(x)$, we have:

The graph of $h$ is a straight line with the slope of 2 and y-intercept at $c = 258$. Ensure it intersects with $f$ at the point $(5, 268)$ and indicate intercepts clearly.

The function p(x) = -rac{8}{x} + 4 has a domain of all real numbers except where the function is undefined, which is at $x = 0$.

Thus, the domain is:

(- ext{∞}, 0) igcup (0, ext{∞})

The function $g(y) = a^2 + q$ is constant with respect to $y$, thus the range is determined by the values of $a$ and $q$. Assuming $a$ is non-certain:

The general range can be expressed as $(- ext{∞}, ext{∞})$ depending on $a$ and $q$.

From previous calculations or given constants, assuming $q = 4$ that was inferred or calculated from the graph context.

Locate $D$, the $y$-intercept of $p$. Setting $x=0$ in $p(x)$, noting $p(x)$ diverges avoids definition at $x=0$, thus: Again, it's often calculated at $y=4$ from the given function, which would imply it does not intersect the y-axis to fall within bounds.

Identify $C$ on the graph as the $y$-intercept: Set $p(0)$ to compute. Eventually, coordinates can derive as the relationship among the plotted lines, intersecting with the $y$-axis.

Setting limits or values found in both expressions might likely give a = rac{1}{2}, assuming value detections lead to resultant simplifications.