IB MYP 4-5 Maths- Representation and shape of more complex functions - Study Notes - New Syllabus

IB MYP 4-5 Maths- Representation and shape of more complex functions – Study Notes

Extended

Representation and shape of more complex functions

IB MYP 4-5 Maths- Representation and shape of more complex functions – Study Notes – All topics

Representation and Shape of More Complex Functions

Functions can be represented in various forms: equations, tables, graphs, and mappings. Different functions have different shapes and properties.

1. Quadratic Functions

General form: \( y = ax^2 + bx + c \), where \( a \neq 0 \).
Graph: A parabola (U-shape).
If \( a > 0 \), parabola opens upwards; if \( a < 0 \), opens downwards.
Vertex: \( \left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right) \right) \).
Axis of symmetry: \( x = -\frac{b}{2a} \).
Domain: \( \mathbb{R} \); Range: \( y \ge \text{vertex y} \) (if \( a > 0 \)), or \( y \le \text{vertex y} \) (if \( a < 0 \)).

Example:

Sketch \( y = x^2 – 4x + 3 \).

▶️ Answer/Explanation

Vertex: \( x = \frac{4}{2} = 2 \), \( y = 2^2 – 4(2) + 3 = -1 \).

So vertex is (2, -1), axis of symmetry x = 2.

Find intercepts: y-intercept = 3, roots = (1, 3).

Graph: U-shape opening upwards.

2. Cubic Functions

General form: \( y = ax^3 + bx^2 + cx + d \).
Graph: S-shaped curve, can have 1 or 2 turning points.
If \( a > 0 \), rises to the right; if \( a < 0 \), falls to the right.
Domain and range: \( \mathbb{R} \).

Example:

Sketch \( y = x^3 – 3x \).

▶️ Answer/Explanation

Find critical points by derivative: \( y’ = 3x^2 – 3 = 0 \) → \( x^2 = 1 \) → x = ±1.

Turning points at (-1, 2) and (1, -2).

Graph passes through origin and has S-shape.

3. Reciprocal Functions

Form: \( y = \frac{k}{x} \), \( x \neq 0 \).
Graph: Two branches (hyperbola) in opposite quadrants.
Asymptotes: x = 0 (vertical), y = 0 (horizontal).
Domain: \( x \neq 0 \); Range: \( y \neq 0 \).

Example:

Sketch \( y = \frac{6}{x} \).

▶️ Answer/Explanation

Two branches in quadrants I and III since k > 0.

Asymptotes: x = 0, y = 0.

4. Exponential Functions

Form: \( y = a^x \), where \( a > 0 \) and \( a \neq 1 \).
If \( a > 1 \), function increases (growth); if \( 0 < a < 1 \), function decreases (decay).
Domain: \( \mathbb{R} \); Range: \( y > 0 \).
Asymptote: y = 0.

Example:

Sketch \( y = 2^x \).

▶️ Answer/Explanation

Curve passes through (0,1), increasing rapidly as x → ∞, approaches 0 as x → -∞.