Additional Mathematics - Graphs

Power Functions

Graphs of Power Functions ($y = ax^n$)

Even Powers: $n = 2, 4, 6, 8, ...$

Even powers create U-shaped or inverted U-shaped curves. When $a > 0$, the parabola opens upward; when $a < 0$, it opens downward.

Odd Powers: $n = 3, 5, 7, 9, ...$

Odd powers create S-shaped curves that pass through the origin. The curves have rotational symmetry about the origin.

Negative Even Powers: $n = -2, -4, -6, -8, ...$

Negative even powers create hyperbolas with vertical asymptote at x = 0. Both branches are in the same vertical half-plane.

Negative Odd Powers: $n = -1, -3, -5, -7, ...$

Negative odd powers create hyperbolas with both vertical and horizontal asymptotes. The branches are in opposite quadrants.

Fractional Powers (x > 0, 0 < n < 1): $y = ax^{1/2}, y = ax^{3/4}, ...$

Fractional powers (0 < n < 1) create root functions that start at the origin and curve upward with decreasing slope.

Exponential & Logarithmic Functions

Exponential Functions ($y = e^x, a^x$ where $a > 1$) and Logarithmic Functions ($y = \ln x, \log_a x$ where $a > 1$)

Exponential and logarithmic functions are inverse functions, reflected across the line y = x. The exponential function grows rapidly for large x, while the logarithmic function grows slowly.

Functions with base 0 < a < 1: $y = e^{-x}, (\frac{1}{2})^x, \log_{0.5} x$

When 0 < a < 1, exponential functions decay as x increases, while their inverse logarithmic functions decrease as x increases.

Modulus Functions

Effect of Modulus on Function Graphs

Original function f(x)

Modulus function |f(x)| - all negative portions are reflected above the x-axis

Key Property of Modulus Functions

When applying the modulus function to any graph y = f(x), all portions below the x-axis (where f(x) < 0) are reflected upward across the x-axis to become positive, while portions above the x-axis remain unchanged.

Trigonometric Graphs

Sine Function: $y = \sin x$

Amplitude: 1
Period: $2\pi$
Domain: All real numbers
Range: [-1, 1]

Cosine Function: $y = \cos x$

Amplitude: 1
Period: $2\pi$
Domain: All real numbers
Range: [-1, 1]

Tangent Function: $y = \tan x$

Period: $\pi$
Domain: All real numbers except $x = \frac{\pi}{2} + n\pi$ where $n$ is an integer
Range: All real numbers
Vertical asymptotes: $x = \frac{\pi}{2} + n\pi$

Parabolas with Horizontal Axis

Graphs of $y^2 = kx$

Horizontal parabolas: When k > 0, the parabola opens to the right; when k < 0, it opens to the left. The vertex is always at the origin.

Circle Properties and Theorems

Key Circle Theorems

1. Tangent Perpendicular to Radius

A tangent to a circle is perpendicular to the radius at the point of tangency.

2. Right Angle in Semicircle

Any angle inscribed in a semicircle is a right angle (90°).

3. Angles in Same Segment

Angles subtended by the same chord in the same segment of a circle are equal.

4. Angle at Centre = 2 × Angle at Circumference

The angle subtended at the centre is twice the angle subtended at any point on the circumference by the same chord.

5. Opposite Angles of Cyclic Quadrilateral

Opposite angles of a cyclic quadrilateral are supplementary: a + c = 180°, b + d = 180°

6. Tangent-Chord Theorem (Alternate Segment Theorem)

If DE is tangent to the circle at B, then ∠CAB = ∠CBE and ∠ACB = ∠ABD (alternate segment theorem).

Congruent and Similar Triangles

Congruent Triangles	Similar Triangles
SSS, SAS, AAS, RHS	SSS, SAS, AAA

Midpoint Theorem

If D and E are midpoints of AB and AC respectively, then DE ∥ BC and DE = ½BC