Graphs
A graph turns a function into a picture — every input-output pair becomes a point on a grid, and a linear function''''s points form a straight line.
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. Try it yourself with input 3: go up from 3, meet the line at height 6, and there is your output.
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. Now when you see any graph, you can read it in two moves: find the input across, follow it up to the curve or line, and read the output on the side.
Practice
Start at the top. Tasks go easiest → hardest: recall a fact, apply it to a case, then a senior-level stretch. Open one, attempt it, then reveal.
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