Linear relations: cheat sheet

Linear relations · The words and methods from the lessons, with an example of each.

Rules, tables and graphs

growing pattern

What each new figure adds goes in front of $n$ ; what is there at the start goes on its own. Figure 4 has $3 \times 4 + 1 = 13$ sticks.

s = 3n + 1

Figure 4: each square adds 3 sticks, plus 1 at the start

ordered pair

Input first, then output. To plot it: across 2 on the $x$ -axis, then up 5.

(x, y) = (2, 5)

rule

A rule turns each input $x$ into an output $y$ . Put the input in for $x$ : $x = 3$ gives $y = 2 \times 3 + 1 = 7$ .

y = 2x + 1

table of values

Choose inputs, work out each output from the rule ( $y = 2x + 1$ here), and write them in pairs.

\begin{array}{c|cccc} x & 0 & 1 & 2 & 3 \\ \hline y & 1 & 3 & 5 & 7 \end{array}

reading a graph

Find the value you know on its axis, go to the line, then read across to the other axis. Sections of a graph increase, decrease or stay constant.

\text{axis} \to \text{line} \to \text{other axis}

vary one variable

To see what one letter does, change only that letter and keep the others the same. Record the results in a table.

V = l \times w \times h \quad (\text{change } h \text{ only})

Lines and their rules

gradient

$m$ , the number in front of $x$ : one step right, the line goes up $m$ (down when $m$ is negative). A bigger $m$ is steeper; $m = 0$ is flat.

y = \mathbf{3}x + 1

vertical and horizontal lines

One letter: every point on $x = 3$ has $x = 3$ (a vertical line); every point on $y = -2$ has $y = -2$ (horizontal).

x = 3 \quad y = -2

y-intercept

$c$ , the number on its own: where the line crosses the $y$ -axis (at $x = 0$ ). Changing $c$ moves the line up or down without changing its steepness.

y = 2x + \mathbf{4}

parallel lines

The same $m$ gives the same steepness, so the lines never meet. $y = 2x + 4$ is $y = 2x + 1$ moved up 3.

y = 2x + 1 \quad y = 2x + 4

y = 2x + 1 and y = 2x + 4: same m, so parallel

inequality region

Draw the boundary $y = 2x + 1$ : dashed for $<$ and $>$ , solid for $\le$ and $\ge$ . Test a point such as $(0, 0)$ : true, shade its side; false, the other side.

y < 2x + 1

y < 2x + 1: below the dashed line, including (0, 0)

Solving and modelling with lines

solve with a graph

Draw each side as a line: $y = 2x + 3$ and $y = 11$ . Where they meet, both sides are equal, so the $x$ there is the solution.

2x + 3 = 11 \;\Rightarrow\; x = 4

rule from a situation

A starting amount plus a rate: the start is $c$ (4 dollars to get in) and the rate is $m$ (2 dollars per km).

T = 2d + 4

comparing two plans

Put both rules on one graph. Where the lines cross, the plans cost the same; on either side, the lower line is cheaper. Check where the model makes sense (no negative gigabytes).

5g + 20 = 40 \text{ at } g = 4