Functions

A function gives one y for each x. Choose the type, enter the numbers and see the graph. Enter an x under "Read off", and the point is drawn so you can read y.

The explanation is shown. Quick: quadratic function and inverse proportion. Advanced: also power and exponential functions, a table of values and what a does to the parabola.

Your numbers

Function

Read off at x =

The point is drawn on the graph.

f(x) = x² − 2x − 3

Answer

Vertex

(1 ; -4)

Zeros

x = -1 and x = 3

Discriminant d

Crosses the y-axis at

(0 ; -3)

f(3)

Key terms: · ·

f(x) = x² − 2x − 3point read off

How to work it out

1f(x) = x² − 2x − 3
2d = b² − 4ac = -2² − 4 · 1 · -3 = 16The discriminant tells you how many times the parabola meets the x-axis: d > 0 twice, d = 0 once, d < 0 never.
3Vertex = (−b / (2a) ; −d / (4a)) = (1 ; -4)a > 0: the arms point up, so the vertex is the lowest point.
4x = (−b ± √d) / (2a) → x = -1 and x = 3The zeros are where the graph crosses the x-axis — there f(x) = 0.
5f(3) = 0

The formula

f(x) = ax² + bx + c

Quadratic function: the graph is a parabola. Vertex (−b/(2a) ; −d/(4a)).

y = k / x

Inverse proportionality: x · y = k.

y = b · xᵃ

Power function: x + p % gives y · (1 + p)ᵃ.

y = b · aˣ

Exponential function: y is multiplied by a each time x rises by 1.

In depth: more numbers and “what if”

What if a is different?

a	Vertex	Zeros
-1	(-1 ; -2)	none
0.5	(2 ; -5)	-1.16 · 5.16
1	(1 ; -4)	-1 · 3
2	(0.5 ; -3.5)	-0.823 · 1.82

Table

x	f(x)
-10	117
-8	77
-6	45
-4	21
-2	5
0	-3
2	-3
4	5
6	21
8	45
10	77

More: Equations and functions

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