Linear relations: recap

A growing pattern adds the same amount each time. Look at what each new figure adds: for a row of squares made of sticks, each new square adds 3 sticks (it shares a side).

So figure $n$ has $n$ lots of 3, plus what every figure has before that: 1 stick on the left. The rule is $s = 3n + 1$.

The rule gives any figure without drawing it: Figure 10 has $3 \times 10 + 1 = 31$ sticks. To go backwards, solve: $3n + 1 = 46$ gives $n = 15$.

A rule like $y = 2x + 3$ turns an input $x$ into an output $y$. To make a table of values, choose some inputs, put each one in place of $x$, and work out its output.

Each input and its output make an ordered pair $(x, y)$: input first, output second. For $x = 4$: $y = 2(4) + 3 = 11$, so the pair is $(4, 11)$, not $(11, 4)$ and not $(4, 4)$.

To go backwards (which input gives an output of 15?), set the rule equal to it and solve: $2x + 3 = 15$, so $x = 6$.

To plot the point $(x, y)$, start at $(0, 0)$, go across $x$ and then up $y$. A negative number goes left or down.

Plot each ordered pair from a rule's table. For $y = 2x + 1$ the pairs $(0, 1)$, $(1, 3)$, $(2, 5)$, $(3, 7)$ lie on a straight line, because each step of 1 across adds the same amount (2) to $y$.

Describe the line by which way it goes and how much for each step: "up 2 for each step right, crossing the $y$-axis at 1". If the steps in the table change, the points curve instead.

A graph of real data pairs two measurements. Check the axis labels first: one measurement is across (the bottom axis) and the other is up the side.

To read a graph, find the value you know on its own axis, go to the line, then read across to the other axis. The point $(15, 24)$ on a graph of temperature against time says: at 15 hours the temperature is 24 °C. Give the units in your answer.

The two numbers of a point belong to different axes. Swapping them gives a different (and wrong) statement.

A graph shows how one measurement changes as another changes. Describe it one section at a time: as time increases, the other measurement increases, decreases or stays constant.

For the bike ride: from 0 to 2 hours the distance from home increases, from 2 to 3 hours it stays constant (the cyclist has stopped), and from 3 to 5 hours it decreases (riding home).

A graph is not a picture of the journey. A line going down does not mean going downhill: it means the measurement on the up-axis is getting smaller.

To see what one variable does in a formula, change that variable only and keep the others the same, then record the results in a table.

In $V = lwh$, doubling any one variable doubles $V$. In $A = s \times s$, doubling $s$ makes $A$ four times as big.

In a rule like $y = 3x + 1$, the number in front of $x$ is called $m$. It tells you how far the line goes up for each step of 1 to the right.

Work it out one step at a time: when $x = 0$, $y = 1$; when $x = 1$, $y = 4$; when $x = 2$, $y = 7$. Each step right, $y$ goes up by 3. That is $m$.

A bigger $m$ makes the line steeper. $m = 0$ makes it flat. A negative $m$ makes it go down. Changing $m$ does not move where the line crosses the $y$-axis: that is the number on its own, $c$.

In $y = mx + c$, $c$ is the number on its own. It is where the line crosses the $y$-axis: when $x = 0$, $mx$ is 0, so $y = c$.

Changing $c$ moves the whole line up or down. It does not change the steepness, because $m$ stays the same: $y = 2x + 4$ is $y = 2x + 1$ moved up 3.

Lines with the same $m$ are parallel: equally steep, so they never meet.

One-letter lines: $x = 3$ is vertical (every point has $x = 3$); $y = -2$ is horizontal.

An inequality such as $y < 2x + 1$ is a region: every point that makes it true. To graph it:

1. Draw the boundary line $y = 2x + 1$: dashed for $<$ and $>$ (its points are not included), solid for $\le$ and $\ge$.
2. Test a point not on the line, often $(0, 0)$: $0 < 1$ is true.
3. Shade the side of the test point if it was true, the other side if it was false.

An equation says two things are equal. Draw each side as a line: for $2x + 3 = 11$, draw $y = 2x + 3$ and $y = 11$.

Where the lines meet, both sides have the same value. The solution is the $x$ of that point: here the lines meet at $(4, 11)$, so $x = 4$. The 11 is the value of both sides, not the answer.

Check by substitution: $2 \times 4 + 3 = 11$. If the lines are parallel, they never meet, and the equation has no solution.

Many real amounts are a starting amount plus a rate. A taxi costs 4 dollars to get in, then 2 dollars for each km: $T = 2d + 4$.

The starting amount is $c$: it is the value when nothing has happened yet (0 km), where the line crosses the axis. The rate is $m$: the words "per", "each" or "every" tell you it is the amount for each one more, how far the line goes up for each step.

The order of the numbers in the words does not matter. Ask: which is paid once (c), and which is paid again and again (m)?

To compare two plans, draw both rules on one graph. Where the lines cross, the plans cost the same.

On each side of the crossing, the lower line is the cheaper plan. A plan with a lower fee is cheaper for small amounts, but a plan with a lower rate wins for large amounts.

Always answer in words ("Up to 4 GB, Plan A is cheaper"), and check the answer makes sense: you cannot use negative data or buy half a ticket, so a model only fits some values.