Functions & Graphing Functions Notes | Junior Cycle Mathematics

A function is a mathematical rule that assigns each input from a domain to exactly one output. In the Junior Cycle Higher Level exam, you will identify patterns from tables, plot linear, quadratic, cubic, and exponential graphs, and use these graphs to solve equations, inequalities, and real-world problems.

Function Concepts, Notation, and Rules

A function takes an input $x$ and applies a rule to produce exactly one output $y$ or $f(x)$ . The vertical line test shows that a curve is a function if every vertical line crosses it at most once.

A vertical line meets an upright parabola once but meets a sideways curve twice, showing two outputs for one input.

The domain is the set of allowed inputs.
The co-domain is the target set that outputs can come from.
The range is the set of actual outputs produced by the domain.

For example, if the domain is $\{-1, 0, 1\}$ and the rule is $x \mapsto x^2$ , the outputs are $0$ and $1$ , giving the range $\{0, 1\}$ .

These notations describe the same rule:

$f(x) = 2x + 3$
$f : x \mapsto 2x + 3$
$y = 2x + 3$

For inputs 0, 1, 2, 3, outputs 1, 4, 9, 16 give $f(x) = (x + 1)^2$ . A €5 fixed charge plus €2 per unit gives $C(x) = 2x + 5$ .

Spotting Patterns in Tables and Graph Shapes

When inputs go up in steps of 1, check the pattern in the outputs:

First differences equal $\rightarrow$ linear.
Constant, non-zero second differences $\rightarrow$ quadratic.
Outputs double or triple for each input step $\rightarrow$ exponential.

Function Type	Form	Graph Shape	Key Features
Linear	$f(x) = mx + c$	Straight line	Constant slope $m$ ; $y$ -intercept at $(0, c)$ .
Quadratic ( $a > 0$ )	$f(x) = ax^2 + bx + c$	U-shaped curve	Symmetrical parabola; lowest point is a minimum turning point.
Quadratic ( $a < 0$ )	$f(x) = ax^2 + bx + c$	$\cap$ -shaped curve	Inverted parabola; highest point is a maximum turning point.
Cubic	$f(x) = x^3 - 4x$	S-shaped curve	Can turn twice; plot from a table of values.
Exponential	$f(x) = 2^x$ or $3^x$	Rising curve	Doubles or triples for each step of 1 in $x$ ; passes through $(0, 1)$ and stays above the $x$ -axis.

Five coordinate sketches show a straight line, upward and downward parabolas, a cubic with two turns, and an increasing exponential curve above the horizontal axis.

Plotting Functions and Domain Rules

Always check the number set specified for the domain before graphing:

Discrete domains ( $x \in \mathbb{N}$ or $x \in \mathbb{Z}$ ): Plot separate dots only. Never join them, because non-integer inputs are not permitted.
Continuous domains ( $x \in \mathbb{R}$ ): Plot the points and join them. Use a straight edge for linear functions, or draw a smooth curve for quadratics, cubics, and exponentials.
Curve smoothly through the lowest or highest point; never draw it flat or make it a sharp corner.
Stay inside the domain given in the question; never extend your graph beyond its stated boundaries.

The rule y=2x+3 appears as three separate points for integer inputs from −1 to 1 and as a bounded straight segment for real inputs over the same interval.

Solving Graphically and Interpreting Real-World Graphs

Graphs provide visual estimates to equations and inequalities:

A root is where $f(x) = 0$ , seen as an $x$ -intercept.
To solve $f(x) = k$ , read the $x$ -coordinates where the horizontal line $y = k$ meets the curve.
To solve $f(x) = g(x)$ , read the $x$ -coordinates where both graphs meet or touch.
To solve $f(x) < g(x)$ , find the $x$ -values where the graph of $f$ sits strictly below $g$ .

On real-world graphs, the $y$ -intercept is the starting value, and the slope is the rate of change. On a distance–time graph, $\text{speed} = \frac{\text{change in distance}}{\text{change in time}}$ . A steeper slope means a faster speed, while a horizontal line means stopped. Show graphical readings clearly; dotted guide lines can help. Read to the graph's scale and the accuracy requested, and state your answer with units.

Key terms

Function: A mathematical relationship where each input produces exactly one output.
Domain: The complete set of allowed input values for a function.
Co-domain: The stated target set containing all the function's potential outputs.
Range: The set of actual outputs produced when every domain value is put into the function.
Vertical Line Test: A visual test where a graph represents a function if no vertical line intersects it more than once.
Root: An input value where $f(x) = 0$ , corresponding to an $x$ -intercept on a graph.
Turning Point: A point where a curve changes from increasing to decreasing, or from decreasing to increasing.

Check yourself

Can an input $x = 2$ map to two outputs $y = 3$ and $y = 7$ in a function?
No, because a function must assign each input to exactly one output.
The values 2, 6, 12, 20 have first differences 4, 6, 8. What type of pattern is this?
Quadratic, because the second differences are all constant at 2.
For $f(x) = x^2$ with domain $\{-2, 0, 2\}$ , what is the range?
$\{0, 4\}$ , because elements in a set are listed only once.
A harbour's water height $h(t)$ in cm has a minimum at $(2.5, 67.5)$ , where $t$ is hours after noon. Interpret this point.
The lowest water height is $67.5\text{ cm}$ , reached $2.5\text{ hours}$ after noon.

Functions & Graphing Functions