IMAT Mathematics · Functions

Graphs of Standard Functions

Lines, parabolas, powers, reciprocals, exponentials and logarithms each have a shape you should recognise, with known intercepts, symmetry and range.

All Functions notes

Why this matters

Quadratic graphs came up in 2020 (the turning point) and 2026 (a parabola through a given point), and 2026 also asked for the values of x where an exponential function is positive.

Concept 1 of 3: Linear and quadratic graphs: gradient, vertex and roots

A linear function grows by the same amount for each step in x, so its graph is a straight line. A quadratic is symmetric about a vertical line, its axis, and turns round at the vertex. Find the axis and you have the vertex; substitute any point on the graph and you get an equation for an unknown coefficient.

Definition

Linear: $y = mx + c$ is a straight line with gradient $m$ and y-intercept $c$ . Quadratic: $y = ax^2 + bx + c$ with $a \ne 0$ is a parabola.

If $a > 0$ it opens upwards and the vertex is a minimum; if $a < 0$ it opens downwards and the vertex is a maximum.
The axis of symmetry is $x = -\dfrac{b}{2a}$ ; the vertex (turning point) is on it.
The y-intercept is $c$ . The x-intercepts (roots) solve $ax^2 + bx + c = 0$ ; there are 2, 1 or 0 of them as the discriminant $b^2 - 4ac$ is positive, zero or negative.
A point $(p, q)$ is on a graph exactly when $q = f(p)$ .

Vertex of y = ax² + bx + c

x_v = -\frac{b}{2a}, \qquad y_v = a x_v^2 + b x_v + c

$x_v$ x-coordinate of the vertex (the axis of symmetry)
$y_v$ y-coordinate of the vertex: substitute $x_v$

Worked example

Find the vertex of

y = 3x^2 - 12x + 7

, say whether it is a maximum or a minimum, and give the y-intercept.

Practice this conceptself-check · 4 quick reps

The axis is at minus b over 2a

The vertex is at

x = -b/(2a)

. Dropping the minus sign puts the vertex on the wrong side of the y-axis, and forgetting the 2 doubles its distance from the axis. Both wrong points appear as options.

Concept 2 of 3: Shapes of power, reciprocal, exponential and logarithmic graphs

You do not need to plot these graphs: learn where each one is defined, which values it can take and one or two points it always passes through. Exponentials and logarithms are mirror images of each other, which is why their domains and ranges swap.

Definition

Key words for reading the table:

An asymptote is a line the graph gets closer and closer to but never reaches.
$e \approx 2.718$ ; $\ln x$ means $\log_e x$ .
For a base $a$ between 0 and 1, $a^x$ and $\log_a x$ are decreasing instead of increasing; the domains and ranges stay the same.

Function	Domain	Range	Shape and key points
$y = x^2$ (even powers)	All real x	$y \ge 0$	U shape, symmetric about the y-axis, through $(0, 0)$ and $(1, 1)$
$y = x^3$ (odd powers)	All real x	All real y	Rises from left to right, symmetric about the origin, through $(0, 0)$ and $(1, 1)$
$y = \sqrt{x}$	$x \ge 0$	$y \ge 0$	Starts at $(0, 0)$ and rises ever more slowly
$y = \dfrac{1}{x}$	$x \ne 0$	$y \ne 0$	Two separate branches; both axes are asymptotes
$y = a^x$ with $a > 1$ , e.g. $e^x$	All real x	$y > 0$	Through $(0, 1)$ ; increasing; the x-axis is an asymptote on the left An exponential is positive for every real x, including negative x.
$y = \log_a x$ with $a > 1$ , e.g. $\ln x$	$x > 0$	All real y	Through $(1, 0)$ ; increasing; the y-axis is an asymptote A logarithm is negative for $0 < x < 1$ and positive for $x > 1$ .

Exponential and logarithm with the same base are inverse functions, so their graphs are reflections of each other in the line

y = x

Practice this conceptself-check · 4 quick reps

An exponential is never zero or negative

e^x > 0

for every real

x

: at

x = -10

it is tiny but still positive. Options such as "only for

x > 0

" or "only for

x \ge 1

" confuse

e^x > 0

with

e^x > 1

, which holds only for

x > 0

Concept 3 of 3: Even and odd functions, increasing and decreasing, reading a graph

Symmetry saves work. An even function looks the same in a mirror held along the y-axis; an odd function looks the same after a half turn about the origin. Most functions are neither. Reading a graph means turning its picture into statements: where it crosses the axes, where it turns, where it is above or below the x-axis.

Definition

Even: $f(-x) = f(x)$ for all $x$ ; the graph is symmetric about the y-axis (e.g. $x^2$ , $x^4 - 3$ ).
Odd: $f(-x) = -f(x)$ for all $x$ ; the graph is unchanged by a half turn about the origin (e.g. $x^3$ , $1/x$ ).
$f$ is increasing on an interval if a larger $x$ always gives a larger $f(x)$ ; decreasing if it gives a smaller one.
On a graph: x-intercepts are the solutions of $f(x) = 0$ , the y-intercept is $f(0)$ , $f(x) > 0$ where the curve is above the x-axis, and turning points are local maxima or minima.

Even and odd

\text{even: } f(-x) = f(x) \qquad \text{odd: } f(-x) = -f(x)

Worked example

Decide whether each function is even, odd or neither:

f(x) = x^4 - 3x^2

g(x) = x^3 + 5x

h(x) = x^2 + x

Practice this conceptself-check · 4 quick reps

A constant term keeps a function even but stops it being odd

Adding a constant to an even function leaves it even (

x^2 + 7

is even). Adding a nonzero constant to an odd function makes it neither (

x^3 + 1

), because an odd function defined at 0 must pass through the origin.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Linear and quadratic graphs: gradient, vertex and roots
Vertex of y = ax² + bx + c
$x_v = -\frac{b}{2a}, \qquad y_v = a x_v^2 + b x_v + c$
Even and odd functions, increasing and decreasing, reading a graph
Even and odd
$\text{even: } f(-x) = f(x) \qquad \text{odd: } f(-x) = -f(x)$

Reference tables (1)

Shapes of power, reciprocal, exponential and logarithmic graphs6 rows

Function	Domain	Range	Shape and key points
$y = x^2$ (even powers)	All real x	$y \ge 0$	U shape, symmetric about the y-axis, through $(0, 0)$ and $(1, 1)$
$y = x^3$ (odd powers)	All real x	All real y	Rises from left to right, symmetric about the origin, through $(0, 0)$ and $(1, 1)$
$y = \sqrt{x}$	$x \ge 0$	$y \ge 0$	Starts at $(0, 0)$ and rises ever more slowly
$y = \dfrac{1}{x}$	$x \ne 0$	$y \ne 0$	Two separate branches; both axes are asymptotes
$y = a^x$ with $a > 1$ , e.g. $e^x$	All real x	$y > 0$	Through $(0, 1)$ ; increasing; the x-axis is an asymptote on the left An exponential is positive for every real x, including negative x.
$y = \log_a x$ with $a > 1$ , e.g. $\ln x$	$x > 0$	All real y	Through $(1, 0)$ ; increasing; the y-axis is an asymptote A logarithm is negative for $0 < x < 1$ and positive for $x > 1$ .

Exponential and logarithm with the same base are inverse functions, so their graphs are reflections of each other in the line

y = x

Watch out for (3)

The axis is at minus b over 2a
→ Linear and quadratic graphs: gradient, vertex and roots
An exponential is never zero or negative
→ Shapes of power, reciprocal, exponential and logarithmic graphs
A constant term keeps a function even but stops it being odd
→ Even and odd functions, increasing and decreasing, reading a graph

Drill the past-year questions on this topic

Open the bank with this topic pre-filtered.

Open in the bank