Introduction to Key Features of Graphs

When analysing a function, we don't just care about its formula , we care about what its graph looks like and what it tells us. The key features of a graph give us a complete picture of a function's behaviour.

In this subtopic, you will learn to identify and interpret:

Maximum and minimum values , the highest and lowest points a function reaches
Intercepts , where the graph crosses the axes
Domain and range , the set of valid inputs and possible outputs, readable directly from a graph
Symmetry , whether a graph has a mirror or rotational pattern
Asymptotes , lines the graph approaches but never touches
Intersections , where two graphs meet

For Math AI SL, technology (graphing calculators or tools like Desmos/GeoGebra) is central to finding these features. You are expected to use technology fluently and interpret the results accurately.

Exam Tip

Get comfortable with your graphing calculator before the exam. Know how to find zeros, local extrema, and intersection points using built-in functions , these skills appear regularly in Paper 2.

Domain and Range from a Graph

Domain: The domain of a function is the set of all possible input values ( $x$ -values) for which the function is defined. On a graph, it is read as the full horizontal extent of the curve.

Range: The range of a function is the set of all possible output values ( $y$ -values) that the function actually produces. On a graph, it is read as the full vertical extent of the curve.

Reading domain and range from a graph is a core skill in SL 2.4. Here is how to do it:

Domain: Look at the graph from left to right. What are the smallest and largest $x$ -values shown? Are the endpoints included (solid dot) or excluded (open circle)?
Range: Look at the graph from bottom to top. What are the smallest and largest $y$ -values reached?

Notation: Domain and range are expressed using interval notation or inequalities:

$0 \leq x \leq 10$ or $[0, 10]$ for a closed interval
$x > 0$ or $(0, \infty)$ for an open-ended interval

Example

The graph of $f (x) = x - 1$ starts at the point $(1, 0)$ and extends to the right, rising without bound.

Domain: The function requires $x - 1 \geq 0$ , so $x \geq 1$ . Written as an interval: $[1, \infty)$ .

Range: The square root is always $\geq 0$ , so $f (x) \geq 0$ . Written as an interval: $[0, \infty)$ .

On a calculator: graph the function and observe that the curve begins at $(1, 0)$ and has no upper bound on the $y$ -axis.

Example

A ball's height is modelled by $H (t) = - 4.9 t^{2} + 19.6 t + 2$ for $t \geq 0$ until it hits the ground.

Using a graphing calculator:

The graph starts at $t = 0$ (when the ball is thrown) and ends when $H = 0$ (when it lands), at approximately $t \approx 4.10$ s.
Domain (in context): $0 \leq t \leq 4.10$
The height rises to a maximum of $22$ m, then falls back to zero.
Range (in context): $0 \leq H \leq 22$

Always interpret domain and range in the context of the problem , not all mathematical values may be physically meaningful.

Warning

When a graph has a vertical asymptote, the $x$ -value of that asymptote is excluded from the domain. When a graph has a horizontal asymptote, the $y$ -value is typically excluded from the range (the function never actually reaches it).

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11 more sections in this topic

Introduction to Key Features of Graphs

In this subtopic, you will learn to identify and interpret:

Maximum and minimum values , the highest and lowest points a function reaches
Intercepts , where the graph crosses the axes
Domain and range , the set of valid inputs and possible outputs, readable directly from a graph
Symmetry , whether a graph has a mirror or rotational pattern
Asymptotes , lines the graph approaches but never touches
Intersections , where two graphs meet

Exam Tip

Get comfortable with your graphing calculator before the exam. Know how to find zeros, local extrema, and intersection points using built-in functions , these skills appear regularly in Paper 2.

Domain and Range from a Graph

Domain: The domain of a function is the set of all possible input values ( $x$ -values) for which the function is defined. On a graph, it is read as the full horizontal extent of the curve.

Range: The range of a function is the set of all possible output values ( $y$ -values) that the function actually produces. On a graph, it is read as the full vertical extent of the curve.

Reading domain and range from a graph is a core skill in SL 2.4. Here is how to do it:

Domain: Look at the graph from left to right. What are the smallest and largest $x$ -values shown? Are the endpoints included (solid dot) or excluded (open circle)?
Range: Look at the graph from bottom to top. What are the smallest and largest $y$ -values reached?

Notation: Domain and range are expressed using interval notation or inequalities:

$0 \leq x \leq 10$ or $[0, 10]$ for a closed interval
$x > 0$ or $(0, \infty)$ for an open-ended interval

Example

The graph of $f (x) = x - 1$ starts at the point $(1, 0)$ and extends to the right, rising without bound.

Domain: The function requires $x - 1 \geq 0$ , so $x \geq 1$ . Written as an interval: $[1, \infty)$ .

Range: The square root is always $\geq 0$ , so $f (x) \geq 0$ . Written as an interval: $[0, \infty)$ .

On a calculator: graph the function and observe that the curve begins at $(1, 0)$ and has no upper bound on the $y$ -axis.

Example

A ball's height is modelled by $H (t) = - 4.9 t^{2} + 19.6 t + 2$ for $t \geq 0$ until it hits the ground.

Using a graphing calculator:

The graph starts at $t = 0$ (when the ball is thrown) and ends when $H = 0$ (when it lands), at approximately $t \approx 4.10$ s.
Domain (in context): $0 \leq t \leq 4.10$
The height rises to a maximum of $22$ m, then falls back to zero.
Range (in context): $0 \leq H \leq 22$

Always interpret domain and range in the context of the problem , not all mathematical values may be physically meaningful.

Warning

Free preview

SL 2.4—Features of a graph

Introduction to Key Features of Graphs

Domain and Range from a Graph

11 more sections in this topic

SL 2.4—Features of a graph

Introduction to Key Features of Graphs

Domain and Range from a Graph

11 more sections in this topic