Mathematics from zero
Graphs
A table of inputs and outputs is correct but hard to feel. Place each pair as a dot on a grid and something happens: the dots line up, climb, or curve. The function becomes a shape you can see at a glance. That picture is a graph.
After this lesson you can say what a graph is, read a point as an input-output pair, explain why a linear function graphs as a straight line, and read an output off a graph for a given input.
A graph shows a function as points on a grid. The grid has two directions. The horizontal axis, running across, carries the input. The vertical axis, running up, carries the output. Every input-output pair the function produces becomes one dot placed on this grid.
Each point is written as a pair: (input, output). A point is named by two numbers
in brackets. The first is how far across — the input. The second is how far up — the
output. The point (3, 9) means “input 3 gave output 9”: go 3 across, then 9 up, and
place the dot.
A linear function graphs as a straight line. The picture above is the function
y = 2x. Each step right moves the same distance up — that is what “linear” meant —
so the dots never bend. They fall exactly on one straight line. Seeing a straight line
is how you recognise a linear function at a glance.
To read an output from a graph, start at the input and go up to the line. Pick a value on the horizontal axis — say 4. Move straight up from it until you meet the line. Then look across to the vertical axis: that height is the output. On the graph above, input 4 meets the line at height 8 — so the function gives 8.
Plot the function y = x + 2 and check that its points form a straight line.
Build a few input-output pairs. Input 0: 0 + 2 = 2, point (0, 2). Input 1:
1 + 2 = 3, point (1, 3). Input 2: point (2, 4). Input 3: point (3, 5).
Place those dots: (0,2), (1,3), (2,4), (3,5). Each step right moves exactly 1
step up — the same step every time.
Because the step never changes, the dots line up perfectly straight. y = x + 2 is
linear, so its graph is a straight line — as expected.
Why this works
Why does a constant step force a straight line and nothing else? Picture walking the graph left to right. At every input you take one step right and the same rise upward. Same move, repeated — your path cannot curve, because curving would need the rise to change. A changing rise bends the graph; a constant rise can only draw a line.
Common mistake
A common mistake is reading a point in the wrong order — taking (3, 7) as “input 7,
output 3”. The order is fixed: first number across (input), second number up (output).
(3, 7) is input 3, output 7. Always read the pair left number first, as how-far-
across then how-far-up.
On the graph of y = 2x, what output does the input 4 give? Type it.
For the point (3, 7), what is the input? Type it.
For the point (3, 7), what is the output? Type it.
On the graph of y = x + 1, what output does the input 5 give? Type it.
On the graph of y = 2x, what output does the input 0 give? Type it.
What does the point (3, 7) on a function's graph mean?
A graph shows a function as points on a grid: the horizontal axis carries the input, the vertical axis the output. Each point is a pair (input, output) — across first, up second. A linear function graphs as a straight line, because its constant step cannot bend. To read an output, go up from the input until you meet the line, then across to the output.