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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

Graph of a Function

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

Let \( f(x) = x^2 \).

▶️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

You may be given a function, a table of values, or a context (word problem). Your task is to sketch an accurate graph based on that data.

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

Given \( f(x) = \frac{1}{x} \), sketch the graph.

▶️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:

Graph Insight: The sum creates a new parabola shifted to the left. 

Example

Sketch the graph of \( f(x) = -x^2 + 4 \) and label all key features.

▶️Answer/Explanation

Solution:

 Vertex at (0, 4)
 x-intercepts at \( x = -2, 2 \)
 y-intercept at \( y = 4 \)
 Opens downward 

Example

Use technology to graph \( f(x) = \frac{1}{x} \) and \( g(x) = \frac{1}{x+2} \).
Describe the transformation.

▶️Answer/Explanation

Solution:

 \( g(x) \) is a horizontal shift of \( f(x) \) left by 2 units.

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