Linear Functions and Graphs

The plane splits into four quadrants, labelled anticlockwise starting at the top right.

A point like $(3, 2)$ sits in the first quadrant, where $x$ and $y$ are both positive. A point like $(-4, 1)$ sits in the second quadrant (negative $x$, positive $y$). The order matters: $(3, 2)$ and $(2, 3)$ are different points. To plot, go along the corridor first, then up the stairs — never the other way around.

• $x$ is the horizontal distance from the origin, $y$ is the vertical.
• Quadrants are numbered Q1, Q2, Q3, Q4 anticlockwise from the top right.
• Order matters: $(3, 2)$ and $(2, 3)$ are different points.
• Plot the $x$ first, then move up or down to the $y$.

For $y = 2x + 1$ the table is:

So plot $(-2, -3)$, $(-1, -1)$, $(0, 1)$, $(1, 3)$, $(2, 5)$ and rule a line through them.

If a single point looks off the line, treat it as a warning: re-check that one calculation rather than forcing the ruler through it. A straight-line graph should never need "best-fit" — every correct point sits on the line.

For special cases, lines like $y = 4$ are horizontal (every point has $y = 4$), and lines like $x = -2$ are vertical (every point has $x = -2$). These are quick to draw — no table needed.

• Use a table of values with at least three rows.
• Substitute each $x$ into the equation to find $y$.
• Plot with care, then draw one straight line through every point.
• Horizontal lines have equation $y =$ constant; vertical lines have $x =$ constant.

A positive gradient slopes up to the right; a negative gradient slopes down; a gradient of zero gives a horizontal line.

The y-intercept, written $c$, is the $y$-value of the point where the line crosses the y-axis (so $x = 0$). For $y = 2x + 1$, set $x = 0$ to get $y = 1$, so the y-intercept is $1$ — the line crosses at $(0, 1)$.

Picking two clean points helps. For a line through $(2, 7)$ and $(5, 13)$ the gradient is $\dfrac{13 - 7}{5 - 2} = \dfrac{6}{3} = 2$. Choose points where the line passes exactly through a grid intersection — your numbers will be cleaner and the answer easier to check.

• Gradient $m = (y_2 - y_1) / (x_2 - x_1)$ — rise over run.
• Positive gradient slopes up, negative slopes down, zero is flat.
• The y-intercept is the $y$-value when $x = 0$.
• Pick points on grid intersections to keep numbers tidy.

• $m$ is the gradient.
• $c$ is the y-intercept.

So $y = 3x - 2$ has gradient $3$ and crosses the y-axis at $-2$. The line $y = -x + 5$ has gradient $-1$ (slopes down) and crosses at $5$.

This makes a lot of questions painless.

• Sketch a line from an equation — mark the y-intercept first, then use the gradient as rise-over-run to step to another point.
• Find the equation from a graph — read $c$ off the y-axis, then use any two clean grid points to find $m$.
• Check whether a point lies on a line — substitute the coordinates into the equation. If both sides match, the point lies on the line.

For example, does $(4, 10)$ lie on $y = 3x - 2$? Substitute: $3(4) - 2 = 10$. Yes! The point lies on the line.

Sometimes the equation is given in a different form, like $2x + y = 7$. Rearrange to make $y$ the subject: $y = -2x + 7$. Gradient $-2$, y-intercept $7$. The form $y = mx + c$ is just one rearrangement away.

• In $y = mx + c$, $m$ is the gradient and $c$ is the y-intercept.
• To sketch, mark $c$ first, then step using rise over run.
• To find the equation, read $c$ off the graph and compute $m$.
• Rearrange equations like $2x + y = 7$ to get $y = -2x + 7$.

If you need a line parallel to $y = 2x + 1$ passing through $(0, -3)$: keep $m = 2$, change $c$ to $-3$, so $y = 2x - 3$.

A few special lines are worth memorising:

• $y = c$ is horizontal — gradient $0$, crosses the y-axis at $c$.
• $x = a$ is vertical — gradient is undefined, crosses the x-axis at $a$.
• $y = x$ is the diagonal through the origin — gradient $1$, y-intercept $0$.
• $y = -x$ is the other diagonal — gradient $-1$, y-intercept $0$.

For a vertical line, the gradient is undefined, not zero — you would be dividing by zero (no run). That's a small but important distinction worth knowing by heart.

• Parallel lines have the same $m$ but different $c$.
• $y = c$ is a horizontal line — gradient $0$.
• $x = a$ is a vertical line — gradient undefined.
• $y = x$ and $y = -x$ are the two main diagonals through the origin.

For a phone plan that charges \3$plus$$2$per minute, the cost$C$in dollars after$m$minutes is$C = 2m + 3$. A graph of$C$against$m$is a straight line with gradient$2$($$2$per minute) and y-intercept$3$(the$$3$fixed cost). Reading the graph, you can answer questions like *"How many minutes for$$25$?"* by setting$C = 25$:$25 = 2m + 3$, so$m = 11$ minutes.

Always label the axes carefully, use the scale the question provides, and pay attention to the units of the gradient.

• Gradient = rate of change (per hour, per item, per minute).
• y-intercept = starting value when $x = 0$.
• Use the graph to predict or compare situations.
• Always check the axes' scales and units before reading off values.

• Simultaneous equations correspond to intersections of straight lines.
• Inequalities use straight lines as boundaries to be shaded.
• Quadratic graphs build on the same coordinate skills.
• Science best-fit lines reuse gradient (rate) and intercept (start).