A Library of Graphs

In short: The main graphs to know by shape are the line, parabola, cubic, hyperbola, exponential and circle. A hyperbola $y = \frac{k}{x}$ has asymptotes it never touches, and the circle with centre $(h, k)$ and radius $r$ is $(x - h)^{2} + (y - k)^{2} = r^{2}$ .

Starting point: Years 9 and 10

Equation	Name	Shape
$y = m x + c$	linear	straight line
$y = x^{2}$	parabola	U shape
$y = x^{3}$	cubic	S shape through the origin
$y = \frac{k}{x}$	hyperbola	two curves that never touch the axes
$y = 2^{x}$	exponential	flat on the left, shoots up on the right
$x^{2} + y^{2} = r^{2}$	circle	centre $(0, 0)$ , radius $r$

Four shapes to know on sight.

Asymptotes

An asymptote is a line a graph gets closer and closer to but never reaches.

$y = \frac{1}{x}$ has asymptotes $x = 0$ and $y = 0$ , because you cannot divide by zero and $\frac{1}{x}$ is never zero.
$y = 2^{x}$ has the asymptote $y = 0$ on the left.

Hyperbolas and inverse variation

$y = \frac{k}{x}$ describes inverse variation: double $x$ and $y$ halves. If $k > 0$ , the curves sit in the first and third quarters. If $k < 0$ , they sit in the second and fourth.

Circles

$x^{2} + y^{2} = r^{2}$ is a circle centred at the origin with radius $r$ . So $x^{2} + y^{2} = 25$ has radius 5.

Graph library

Polynomials

Line, parabola, cubic
Roots from factors

Hyperbola

y = k/x
Two asymptotes

Exponential

y = aˣ
Asymptote y = 0

Circle

x² + y² = r²
Centre (h, k) form

Absolute value

V shape

Year 11Shifted hyperbolas and absolute value

$y = \frac{a}{x - b} + c$ is a hyperbola with vertical asymptote $x = b$ and horizontal asymptote $y = c$ .

$y = ∣ x ∣$ is a V shape with its corner at the origin, because $∣ x ∣ = x$ when $x \geq 0$ and $∣ x ∣ = - x$ when $x < 0$ . The graph of $y = ∣ x - 2∣ + 1$ has its corner at $(2, 1)$ .

To solve $∣2 x - 1∣ = 5$ , split it: $2 x - 1 = 5$ or $2 x - 1 = - 5$ , so $x = 3$ or $x = - 2$ .

Horizontal asymptotes of fractions. For a fraction of polynomials, look at large $x$ . If the top and bottom have the same highest power, the asymptote is the ratio of their leading numbers: $y = \frac{2 x ^{2} + x}{x ^{2} + 1}$ approaches $y = 2$ . If the bottom has the higher power, the asymptote is $y = 0$ . Vertical asymptotes are where the bottom is zero (and the top is not).

Shifted exponentials. $y = a \times 2^{x} + c$ has horizontal asymptote $y = c$ and $y$ -intercept $a + c$ . So an asymptote at $y = 1$ and $y$ -intercept 4 give $c = 1$ and $a = 3$ .

Counting solutions with graphs. To find how many solutions $x^{2} = 2^{x}$ (or any hard equation) has, sketch $y = x^{2}$ and $y = 2^{x}$ on the same axes and count the crossings. Moving one side across the equals sign is fine: $e^{x} + x - 2 = 0$ has as many solutions as $e^{x} = 2 - x$ has crossings (here, one).

Nearest and farthest points on a circle. They lie on the line through the point and the centre. The origin is 5 units from the centre $(3, 4)$ of a circle of radius 2, so the nearest point is 3 units away and the farthest is 7 units away, both on the line $y = \frac{4}{3} x$ .

Quick summary

Know the shapes: line, parabola, cubic, hyperbola, exponential, circle.
An asymptote is a line the graph approaches but never touches.
$y = \frac{k}{x}$ : inverse variation with asymptotes on both axes.
$x^{2} + y^{2} = r^{2}$ is a circle of radius $r$ centred at the origin.
Circle with centre $(h, k)$ : $(x - h)^{2} + (y - k)^{2} = r^{2}$ .

Quiz

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Try 3 questions now

Which equation gives an S-shaped curve through the origin?
1. A $y = x^{2}$
2. B $y = x^{3}$
3. C $y = \frac{1}{x}$
4. D $y = 2^{x}$
Show answer
B: $y = x^{3}$
The basic cubic.
What is the radius of the circle $x^{2} + y^{2} = 49$ ?
1. A 49
2. B 24.5
3. C 7
4. D 14
Show answer
C: 7
$r^{2} = 49$ , so $r = 7$ .
What are the asymptotes of $y = \frac{4}{x}$ ?
1. A $x = 4$ and $y = 0$
2. B $x = 0$ and $y = 0$
3. C $y = 4$ only
4. D it has none
Show answer
B: $x = 0$ and $y = 0$
You cannot divide by zero, and the output is never zero.