Linear functions and Graphs

The golden rule for reading coordinates is across before up — the $x$-number first, the $y$-number second. A useful memory hook: "along the corridor, then up the stairs".

The two axes cut the grid into four quadrants. In some of them, $x$ or $y$ (or both) are negative. So a point can be $(3, 2)$, but it can also be $(-2, -3)$ or $(-4, 1)$. The numbers tell you exactly where to go: a negative $x$ means go left, a negative $y$ means go down.

• Coordinates are written $(x, y)$ — across, then up.
• The axes are the $x$-axis and $y$-axis; they cross at the origin.
• The origin is the point $(0, 0)$.
• Four quadrants allow negative $x$ and $y$ values.

To read a point that is already plotted, do the same in reverse: see how far it is along the $x$-axis, then how far up the $y$-axis.

Accuracy matters. A point plotted one square out will pull your whole graph out of shape. Take your time, count carefully, and keep your dots small so they sit exactly on the grid line crossings.

• Always start counting from the origin.
• Move across for $x$ first, then up or down for $y$.
• Negative numbers mean go left or down.
• Keep marks small and neat for accuracy.

We often write a function as an equation. If the input is $x$ and the output is $y$, the machine above is $y = 2x + 1$.

The one strict rule of a function: each input gives exactly one output. You can never put 4 in and get two different answers out.

A function is also called a mapping, because it maps (sends) each input to its matching output. Functions, mappings and number machines are three names for the same idea.

• A function turns an input into one output, by a rule.
• Picture a number machine: input → rule → output.
• $y = 2x + 1$ means output $= 2 \times$ input $+ 1$.
• Each input gives exactly one output.

Take the function $y = 2x + 1$:

To fill the table:

• When $x = 0$: $y = 2 \times 0 + 1 = 1$.
• When $x = 1$: $y = 2 \times 1 + 1 = 3$.
• When $x = 2$: $y = 2 \times 2 + 1 = 5$.
• When $x = 3$: $y = 2 \times 3 + 1 = 7$.

Notice how $y$ goes up by 2 each time $x$ goes up by 1 — that steady jump is a sign the points will form a straight line. A neat table makes plotting quick and almost mistake-proof.

• A table of values lists inputs and their outputs.
• Run each $x$ through the rule to find $y$.
• Each column is a coordinate pair $(x, y)$.
• A steady jump in $y$ hints at a straight line.

A neatly drawn graph is a tool you can read in two directions:

• From $x$ to $y$: pick a value on the $x$-axis, go straight up to the line, then across to the $y$-axis.
• From $y$ to $x$: start on the $y$-axis, go across to the line, then down to the $x$-axis.

This is why graphs are so useful — you can find an answer without doing the calculation. If your points do not line up, that is a clear signal to check your table: a straight-line function must give points in a straight row.

• Plot every point from the table.
• Join them with a ruler — a linear function gives a straight line.
• Read from $x$ to $y$: up to the line, then across.
• Read from $y$ to $x$: across to the line, then down.