5. A Library of Graphs | Mathematics Advanced

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

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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