Key features of graphs and GDC analysis

IB Mathematics Applications and Interpretation SL· 12 min read

1. Core Defined Features of 2D Graphs★★☆☆☆⏱ 10 min

🚫 No Calculator

Every 2D function graph for the AI SL syllabus has a standard set of measurable features that exam questions will ask you to identify or interpret.

📘 Definition

y-intercept

(0, c)

The point where the graph crosses the y-axis, calculated by substituting x=0 into the function

Example:

For f(x) = 2x² + 3x - 5, the y-intercept is (0, -5)

x-intercepts (roots): Values of x where f(x)=0
Local maxima/minima: Turning points where the gradient changes sign
Horizontal asymptotes: Horizontal lines the graph approaches as x → ±∞
Vertical asymptotes: Undefined points where the function value tends to ±∞

📐 Worked Example

List all key features of the graph of $f (x) = \frac{6}{x - 2} + 1$

1
First calculate the y-intercept by substituting x=0
2
$f(0) = \frac{6}{0-2} + 1 = -3 + 1 = -2, \text{ so } (0, -2)$
3
Find the x-intercept by setting f(x)=0
4
$\frac{6}{x-2} + 1 = 0 \implies \frac{6}{x-2} = -1 \implies x - 2 = -6 \implies x = -4$
5
Locate the vertical asymptote where the denominator is zero
6
$x - 2 = 0 \implies x = 2$
7
Identify the horizontal asymptote from the end behaviour as x → ±∞
8
$\frac{6}{x-2} \to 0, \text{ so } y = 1$

2. Step-by-Step GDC Workflow for Extracting Graph Features★★★☆☆⏱ 15 min

📊 Graphing only

IB AI SL Paper 2 questions almost always allow and encourage GDC use to avoid tedious algebraic solving of roots or turning points.

📐 Worked Example

Use your GDC to find the local minimum of f(x) = 0.5x³ - 3x² + 4 on the domain x ∈ [-2, 6]

1
1. Input the full function into your GDC Y= editor, press graph to plot
2
1. Open the 'Calculate' menu, select the 'Minimum' tool
3
1. Select a left bound to the left of the visible low point, a right bound to the right, and press enter to confirm your guess
4
1. Record the output values: x = 4, y = -9.9999 which rounds to exactly (4, -10) for your final answer

3. Contextual Interpretation of Graph Features★★★☆☆⏱ 10 min

Many AI SL graph feature questions are set in real-world contexts, so you will need to translate abstract x/y values into the scenario given.

✓ Quick check

Test your interpretation skills for a population growth model P(t) where t is time in years:

What does the horizontal asymptote of P(t) represent?
- The initial population at t=0
- The maximum sustainable population the environment can support
- The time when the population stops growing
Reveal answer
The maximum sustainable population the environment can support —
This is the carrying capacity, a standard interpretation for logistic growth graphs in AI SL.

4. GDC Accuracy Checks for Full Marks★★★★☆⏱ 8 min

A very common source of lost marks is unrounded or incorrectly rounded GDC outputs, which IB examiners penalize heavily.

Double check that your GDC is set to at least 4 decimal places in the mode menu
If a question specifies a domain, restrict your GDC search to that domain to avoid finding off-range turning points
Always write down the GDC menu tool you used (e.g. 'GDC: minimum function') to show your working even if you do not show algebra

5. Common Pitfalls

Wrong move:

Forgetting to adjust the GDC window, leading you to miss a second x-intercept outside the default view

Why:

Default GDC windows only show x from -10 to 10, which may not cover the full domain specified in the question

Correct move:

Set your Xmin, Xmax, Ymin, Ymax to exactly match the domain and range given in the question before running any calculate tools

Wrong move:

Rounding a GDC turning point value to 2 significant figures instead of 3

Why:

IB AI SL mark schemes require 3 significant figures for all non-exact values unless explicitly stated otherwise

Correct move:

Keep the full unrounded GDC value in your calculator memory, then round the final answer to 3 significant figures before writing it down

Wrong move:

Confusing a vertical asymptote with an x-intercept where the graph dips very close to the x-axis

Why:

GDC trace mode can give you a very small y value that looks like zero, but is not actually a root

Correct move:

Verify any suspected root by substituting the x value back into the original function to confirm f(x) = 0

Wrong move:

Stating that a horizontal asymptote is crossed at some large x value

Why:

Horizontal asymptotes only describe behaviour as x tends to ±infinity, graphs can cross them at finite x values

Correct move:

Only reference horizontal asymptote behaviour for very large positive or negative x values when answering exam questions

Wrong move:

Using algebraic solving for a 4th degree function root instead of using the GDC

Why:

The IB explicitly awards marks for efficient GDC use on Paper 2, and wasting time on algebra will leave you with no time for later questions

Correct move:

If the function is higher than degree 2, use your GDC calculate tool to find roots and turning points immediately

6. Quick Reference Cheatsheet

Graph Feature	GDC Tool Name (TI-84)	Required Rounding
x-intercept / root	Zero	3 s.f.
y-intercept	Value (x=0)	Exact if possible, else 3 s.f.
Local maximum	Maximum	3 s.f.
Local minimum	Minimum	3 s.f.
Intersection of two graphs	Intersect	3 s.f.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · Paper 2
GDC graph feature extraction
2024 · Paper 1
Identify asymptotes of rational function
2023 · Paper 2
Interpret turning point for profit model

Going deeper

gdc_guideTI-84 Plus Graph Feature Step-by-Step WalkthroughOfficial IB-approved GDC workflow for this topic

What's Next

Now that you can reliably extract and interpret all key graph features using your GDC, you are ready to apply these skills to full function modelling questions, which are a major part of the AI SL course. You will practice fitting regression models to real data sets, using your graph feature knowledge to validate if a linear, quadratic, or exponential model is the best fit for the given data. You will also learn how to use GDC graph analysis to solve optimisation problems, where you find the maximum or minimum value of a real world quantity like profit or volume to get full marks on extended response questions.