POP

C2.10 Graphs of Functions

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01 — Explanation

How Do I Turn a Function Into a Table of Values?

Choose several $x$ -values across the given range, substitute each into the function to find the matching $y$ -value, and record the pairs in a table. Every point $(x,y)$ from the table can then be plotted on a grid. Working through the $x$ -values in order, left to right, makes it far less likely that a substitution gets skipped or repeated by mistake.

02 — Explanation

What Do Linear, Quadratic and Reciprocal Graphs Look Like?

Three function families are required at Core level, and each has its own instantly recognisable shape:

Form	Shape	Notes
$y = ax+b$	Straight line	$a$ is the gradient, $b$ the $y$ -intercept
$y = \pm x^2+ax+b$	Parabola	Symmetrical U-shape ( $+x^2$ ) or n-shape ( $-x^2$ )
$y = \dfrac{a}{x}\ (x\neq0)$	Reciprocal curve	Two separate branches; undefined at $x=0$

$y = x^2-2x-3$ for $-2 \leqslant x \leqslant 4$ : substituting $x=-2$ gives $y=(-2)^2-2(-2)-3=5$ , and $x=1$ gives $y=1-2-3=-4$ , filling in the table:

$x$	$-2$	$-1$	$0$	$1$	$2$	$3$	$4$
$y$	$5$	$0$	$-3$	$-4$	$-3$	$0$	$5$

Notice the symmetry in the $y$ -values — this is a useful self-check, since a genuine parabola's $y$ -values should rise and fall in a mirror-image pattern either side of its lowest (or highest) point.

Exam Tip

Straight-line graphs are joined with a ruler; quadratic and reciprocal graphs are joined with a single smooth freehand curve, never point-to-point with a ruler.

03 — Explanation

What Happens to a Reciprocal Graph Near x=0?

$y = \dfrac{a}{x}$ is undefined at $x=0$ , so the graph splits into two separate branches that never touch the $y$ -axis. As $x\to0$ from the positive side, $y$ shoots off towards $+\infty$ ; from the negative side, $y$ shoots off towards $-\infty$ .

For $y=\dfrac{6}{x}$ : at $x=2$ , $y=3$ ; at $x=-3$ , $y=-2$ . As $x$ becomes very large (either positive or negative), $y$ gets closer and closer to $0$ without ever reaching it — both branches of the curve flatten out as they move away from the origin.

04 — Explanation

How Do I Solve an Equation Using My Graph?

The roots of $f(x)=0$ are exactly the $x$ -values where the curve crosses the $x$ -axis — read them straight off the graph or the table.

Solve $x^2-2x-3=0$ using the table above: $y=0$ at $x=-1$ and $x=3$ , so:

x=-1 \quad \text{or} \quad x=3

05 — Explanation

How Do I Use Two Graphs Together to Solve Simultaneous Equations?

Wherever two graphs cross, both equations are true for that same $(x,y)$ pair — so an intersection point is a solution to the pair of equations.

The line $y=x+1$ and the curve $y=x^2-1$ intersect at $(-1,0)$ and $(2,3)$ , giving the solutions:

x=-1,\ y=0 \qquad \text{and} \qquad x=2,\ y=3

Exam Tip

Always read off both coordinates of an intersection point — giving only the $x$ -value when a question asks for the solution to a pair of simultaneous equations loses the accompanying $y$ -value mark.

06 — Explanation

Exam-Style Question

State what happens to the graph of $y=\dfrac{6}{x}$ as $x$ gets very close to $0$ from the negative side.

Solution

As $x\to0^-$ , dividing a fixed positive number ( $6$ ) by a number that's shrinking towards zero but staying negative produces an increasingly large negative result — the branch plunges steeply downward, getting closer and closer to (but never touching) the $y$ -axis.