Mathematics Marking Guideline Analysis - NSC Grade 10 Functions

Determining the Exponential Function $g(x)$ * The function is defined by the general form: $g(x) = b^{x}$ . * To solve for the base $b$ , a point from the graph is substituted into the equation. Using the coordinates $(2; 4)$ , the equation becomes: $4 = b^{2}$ . * Solving for the base: $b = 2$ . * Marking Criteria for 3.1 (Exponential): * One mark for the correct substitution of the point into the equation. * One mark for identifying the correct final value of $b = 2$ .
Determining the Parabolic Function $h(x)$ * The function is defined by the general form: $h(x) = ax^{2} + q$ . * The parameter $q$ represents the vertical shift or the y-intercept of the parabola. From the transcript, it is determined that: $q = -2$ . * To solve for the coefficient $a$ , the point $P(1; 0)$ is substituted into the equation: $0 = a(1)^{2} - 2$ . * Solving for $a$ : $a = 2$ . * Marking Criteria for 3.1 (Parabolic): * One mark for identifying the value of $q = -2$ . * One mark for the correct substitution of point $P$ . * One mark for identifying the correct final value of $a = 2$ .
Coordinates of Point B * The x-intercept of the parabolic function opposite to point $P$ is identified as: $B(-1; 0)$ . * Marking Criteria for 3.2: * One mark for the correct coordinates of point $B$ .

Calculating the Gradient and Equation of a Line (Question 3.3) * The gradient ( $m$ ) of the line is calculated using the formula for slope: $m = \frac{4 - 0}{2 - 1}$ . * Calculation: $m = \frac{4}{1} = 4$ . * The general linear equation is: $y = mx + c$ . * Substituting the gradient ( $m = 4$ ) and the point $(1; 0)$ to find the y-intercept ( $c$ ): * $0 = 4(1) + c$ * $c = -4$ * The final equation of the line is: $y = 4x - 4$ . * Marking Criteria for 3.3: * One mark for the correct calculation of the gradient $m = 4$ . * One mark for the correct value of the y-intercept $c$ . * One mark for providing the final linear equation.
Defining the Range (Question 3.4) * The range of the function is expressed as the set of all possible y-values. * The range is given as: $y \le 2$ . * Alternative interval notation: $y \in (-\infty; 2]$ . * Marking Criteria for 3.4: * One mark for the correct identification of the range limit and inequality/interval.

Sketching and Properties of Hyperbolic Functions (Question 4.1.1) * The marking guidelines specify five key components for the visual representation or analysis of the graph: * Shape: The general curvature of the hyperbola must be correct relative to its quadrants. * Asymptotes: Correct placement of vertical and horizontal lines that the graph approaches but never touches. * X-intercepts: Identification of where the graph crosses the horizontal axis. * Y-intercept: Identification of where the graph crosses the vertical axis. * Numerical values associated with this section include: $1.5$ , $-4.5$ , $3$ , $0.5$ , and $2.5$ . * Marking Criteria for 4.1.1: * One mark for the correct shape. * One mark for the correct asymptotes. * Two marks for the correct x-intercepts. * One mark for the correct y-intercept.
Solving for X in a Rational Equation (Question 4.1.2) * Equation provided: $-2 = -\frac{1}{x} - 1$ . * Step-by-step algebraic solution: * Add 1 to both sides: $-1 = -\frac{1}{x}$ . * Multiply both sides by $-x$ : $-x = 1$ . * Final result: $x = -1$ . * Marking Criteria for 4.1.2: * One mark for the correct final answer of $x = -1$ .

Finding the Constant K (Question 4.2.1) * Based on the provided ratio or substitution: $\frac{k}{2} = 1$ . * Solving for the variable: $k = 2$ . * Marking Criteria for 4.2.1: * One mark for the correct identification of $k = 2$ .