IB Mathematics AI SL The graph of a function MAI Study Notes - New Syllabus

IB Mathematics AI SL The graph of a function MAI Study Notes

LEARNING OBJECTIVE

The graph of a function; its equation y = f(x)

Key Concepts:

Graphing Functions

MAI HL and SL Notes – All topics

The Graph of a Function; Equation $ y = f(x) $

A function relates every input $ x $ to exactly one output $ y $.

It is commonly expressed as $ y = f(x) $, where:

$ x $: independent variable (input),
$ f(x) $: dependent variable (output).

Graphs visually represent how $ f(x) $ changes with $ x $.

Key Concept:

A graph passes the vertical line test if it’s a function.

Example

$ f(x) = x^2 $.

Sketch the graph of given function.

Also discuss about vertex.

▶️Answer/Explanation

Solution:

It’s a quadratic function.

The graph is a parabola opening upwards with vertex at (0, 0).

Sketch description: Smooth U-shaped curve, symmetric about the y-axis.

Sketching a Graph from Information or Context

What to include in your sketch:

Axes with labels and scale.

Important points:

x-intercepts: where $ f(x) = 0 $,
y-intercept: where $ x = 0 $,
Turning points (max/min),
Asymptotes (if applicable),
End behavior (what happens as $ x \to \infty $ or $ -\infty $).

Sketching Tips:

Draw = use scale, ruler, accurate points.
Sketch = rough shape showing correct features.

Example

$ f(x) = \frac{1}{x} $

Sketch the graph of given function.

Also discuss about Asymptotes

▶️Answer/Explanation

Solution:

Asymptotes: vertical at $ x = 0 $, horizontal at $ y = 0 $
No intercepts.
Hyperbolic shape in 1st and 3rd quadrants.

Sketch description: Two curves approaching the axes but never touching.

Using Technology to Graph Functions

Technology (e.g., Desmos, GeoGebra, GDC) is encouraged to:
Plot single functions like $ f(x) = 2x + 3 $
Graph sums/differences: $ f(x) + g(x) $, $ f(x) – g(x) $
Check your sketch against an accurate graph.

Example

Let $ f(x) = x^2 $, $ g(x) = 2x $.

Use a graphing calculator to plot:

$ f(x) $
$ g(x) $
$ f(x) + g(x) = x^2 + 2x $

▶️Answer/Explanation

Solution: