Summary (Grade 11 NSC Matric Mathematics): Revision Notes

Summary

Parabolic functions

Parabolic functions are quadratic functions that create U-shaped or inverted U-shaped graphs called parabolas. These are among the most frequently tested functions in NSC Mathematics.

Standard form

The standard form of a parabolic function is: $y = ax^2 + bx + c$

Key features in standard form:

• y-intercept: $(0; c)$ - where the graph crosses the y-axis
• x-intercept: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ - where the graph crosses the x-axis
• Turning point: $\left(-\frac{b}{2a}; -\frac{b^2}{4a} + c\right)$ - the highest or lowest point on the parabola
• Axis of symmetry: $x = -\frac{b}{2a}$ - the vertical line that divides the parabola into two equal halves

Completed square form

The completed square form is: $y = a(x + p)^2 + q$

This form makes it much easier to identify transformations and the turning point directly from the equation.

Key features in completed square form:

• Turning point: $(-p; q)$ - this form makes it easy to identify the vertex
• Horizontal shifts: $p > 0$ shifts the graph left, $p < 0$ shifts the graph right
• Vertical shifts: $q > 0$ shifts the graph up, $q < 0$ shifts the graph down

Average gradient

The average gradient measures the rate of change between two points on any function. This concept applies to all types of functions, not just linear ones.

Formula: Average gradient $= \frac{y_2 - y_1}{x_2 - x_1}$

This formula calculates the slope of the straight line connecting two points $(x_1; y_1)$ and $(x_2; y_2)$ on a function.

Hyperbolic functions

Hyperbolic functions have the characteristic shape of a hyperbola with vertical and horizontal asymptotes. These functions are undefined at certain x-values where division by zero occurs.

Standard form

$y = \frac{k}{x}$

Key properties:

• When $k > 0$: the graph appears in the first and third quadrants
• When $k < 0$: the graph appears in the second and fourth quadrants

Shifted form

$y = \frac{k}{x+p} + q$

Transformation effects:

• $p > 0$: horizontal shift left
• $p < 0$: horizontal shift right
• $q > 0$: vertical shift up
• $q < 0$: vertical shift down
• Asymptotes: $x = -p$ (vertical) and $y = q$ (horizontal)

Exponential functions

Exponential functions show rapid growth or decay and never touch the x-axis. These functions model many real-world phenomena like population growth and radioactive decay.

Standard form

$y = ab^x$

Key properties:

• When $a > 0$: the graph lies above the x-axis
• When $a < 0$: the graph lies below the x-axis
• When $b > 1$: increasing function if $a > 0$, decreasing function if $a < 0$
• When $0 < b < 1$: decreasing function if $a > 0$, increasing function if $a < 0$

Shifted form

$y = ab^{(x+p)} + q$

Transformation effects:

• $p > 0$: horizontal shift left
• $p < 0$: horizontal shift right
• $q > 0$: vertical shift up
• $q < 0$: vertical shift down
• Asymptote: $y = q$ (horizontal line)

Trigonometric functions

Trigonometric functions are periodic, meaning they repeat their patterns at regular intervals. Understanding their transformations is crucial for NSC Mathematics.

Sine functions

Shifted form: $y = a \sin(k\theta + p) + q$

Key properties:

• Period $= \frac{360°}{|k|}$ - how often the pattern repeats
• $k > 1$ or $k < -1$: period decreases (function oscillates faster)
• $0 < k < 1$ or $-1 < k < 0$: period increases (function oscillates slower)
• $p > 0$: horizontal shift left
• $p < 0$: horizontal shift right
• $q > 0$: vertical shift up
• $q < 0$: vertical shift down

Cosine functions

Shifted form: $y = a \cos(k\theta + p) + q$

Key properties:

• Period $= \frac{360°}{|k|}$ - same period formula as sine
• $k > 1$ or $k < -1$: period decreases
• $0 < k < 1$ or $-1 < k < 0$: period increases
• $p > 0$: horizontal shift left
• $p < 0$: horizontal shift right
• $q > 0$: vertical shift up
• $q < 0$: vertical shift down

Tangent functions

Shifted form: $y = a \tan(k\theta + p) + q$

Key properties:

• Period $= \frac{180°}{|k|}$ - note that tangent has a shorter period than sine and cosine
• $k > 1$ or $k < -1$: period decreases
• $0 < k < 1$ or $-1 < k < 0$: period increases
• $p > 0$: horizontal shift left
• $p < 0$: horizontal shift right
• $q > 0$: vertical shift up
• $q < 0$: vertical shift down
• Asymptotes: $\frac{90° - p}{k} \pm \frac{180°n}{k}$, where $n \in \mathbb{Z}$ (any integer)