A function is a rule: put a number in, get a number out. Draw those answers as points and you get a graph. Pick an input and help Pip collect the points.
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| x (in) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y (out) | 1 | 1.5 | 2 | 2.5 | 3 |
Joining all those points draws a path, so the graph is just the rule drawn as a picture.
A point on a graph is described by two numbers: how far across (x), then how far up (y). Go 2 to the left and 3 up and you are at (−2, 3).
In Pip, he always walks along the ground to the x-value first, and then jumps up for the y-value.
Every spot in the plane has an address of two numbers: how far across, then how far up.
A function takes an input and gives back an output. Put in 3 and out comes 5, every single time.
On a graph, the input is how far across you go, and the output is how far up. Same input, same output: that is what makes it a function.
A rule is a machine: give it the same number and it always gives back the same answer.
Writing f(2) asks the rule for its answer at 2. It is just a short way to say “the output when the input is 2”.
On the graph, f(2) is the height of the path above 2. Go up from 2 on the x-axis until you reach the path, then read across to the y-axis.
Writing f(2) asks the rule for its answer at 2, which on the graph is the height of the path above 2.
A function gives one answer for each x. So its path can never pass over the same x twice.
Draw a straight up-and-down line anywhere. If it meets the path once, that path could be a function. If it meets the path more than once, it is not. This is called the vertical line test.
A rule gives one answer for each x, which is why its path can never pass over the same x twice.
A function is a rule that gives exactly one output for every input. Put in the same number and you always get the same answer back.
f(2) is the output of the function f when the input is 2. On a graph, it is the height of the path above 2 on the x-axis.
Use the vertical line test. Draw a straight up-and-down line anywhere on the graph. If it never meets the path more than once, the graph can be a function. If it meets the path more than once, it is not.
Work out the output for several inputs, mark each pair (x, y) as a point on the grid, and join the points. For y = 0.5x + 1 the points (0, 1), (1, 1.5), (2, 2), (3, 2.5) and (4, 3) all lie on one straight line.
World 0 of Pip is all about functions: coordinates, input and output, tables, graphs and f(x). Pip walks, jumps and climbs your answers, so you see each idea work instead of memorizing it.
It is a math game for iPhone, coming soon to the App Store. Join the list and we will tell you the day Pip is out.