Year 7 Maths Number Plane

Graphing Table of Values

Q: What is a table of values?

A table of x-values and the y-values a rule gives for them. Each column is a point (x, y).

Q: How do you complete a table of values?

Substitute each x-value into the rule. For y = x + 3 and x = 2, y = 5.

Q: How do you find the rule from a table?

Find how much y changes when x goes up 1, multiply x by that step, then add or take away to match the table.

Q: What line do the points (5, 1), (5, 2), (5, 3) make?

The vertical line x = 5, because every x is 5.

Q: How do you check if a point is on a line?

Substitute the x-value into the rule. If the answer is the y-value, the point is on the line.

20 practice questions 1 video lesson Theory + worked examples

Graphing Table of Values links a rule, a table of values and a straight-line graph in NSW Year 7 maths (NESA outcome MA4‑LIN‑C‑01). Each column of the table gives a pair of coordinates, and plotting the points for a rule such as $y = x + 2$ shows them lying on a line.

You will complete tables from a rule, plot and join the points, describe patterns such as rows of matchstick squares with an equation, use a rule to predict larger values, and find where two lines cross (the point of intersection). An electrician's call-out fee plus an hourly rate and a draining water tank are real examples. It is part of Year 7 Number Plane.

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Practice questions

Every question with a fully worked solution.

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Graphing Table of Values - Video - Graphing straight lines using a table of values Watch

Theory

A table of values lists $x$-values and the $y$-values a rule gives for them. Each column is a point $(x, y)$. When the points lie on a straight line, the relationship is linear. This Year 7 maths page shows how to fill in a table, plot the points, find the rule from a table and use the line to answer questions.

Ordered pairs can be written in a table of values. Each column (or row) is one point:

$x$	$-1$	$0$	$2$	$4$
$y$	$0$	$3$	$5$	$1$

means the points $(-1, 0)$, $(0, 3)$, $(2, 5)$, $(4, 1)$. A table can also go down the page, with an $x$ column and a $y$ column.

Rule to table. Substitute each $x$-value into the rule. For $y = 2x - 1$ and $x = 4$: $y = 2 \times 4 - 1 = 7$.

Table to graph. Plot each point, then join them with a ruler and put arrows on both ends: the line keeps going, so infinitely many points fit the rule. Points that lie on one straight line are collinear.

Special lines. If every $y$-value is $2$, the points make the horizontal line $y = 2$. If every $x$-value is $2$, they make the vertical line $x = 2$.

For $y = 2x - 1$: $x = 0, 1, 2, 3$ gives $y = -1, 1, 3, 5$. Each time $x$ goes up $1$, $y$ goes up $2$. Extending the line finds more points, such as $(-1, -3)$: $2 \times (-1) - 1 = -3$.

Every point on the red line has $y = 3$: it is the horizontal line $y = 3$. Every point on the blue line has $x = 1$: it is the vertical line $x = 1$.

From a table to a rule:

Going up. If $y$ goes up by the same step each time $x$ goes up $1$, multiply $x$ by the step, then add or take away to match the table.

$x$	$0$	$1$	$2$	$3$
$y$	$-1$	$1$	$3$	$5$

$y$ goes up $2$ each time, so start with $2x$. At $x = 0$, $2x = 0$ but $y = -1$, so take away $1$:

\[ y = 2x - 1 \]

y = 2 x - 1

Going down. If $y$ goes down by the same step, start with the $y$-value at $x = 0$ and take away the step for each $x$: $y = 10, 7, 4, 1$ for $x = 0, 1, 2, 3$ gives $y = 10 - 3x$.

          Is a point on the line? Substitute its \(x\)-value into the rule. If you get its \(y\)-value, the point is on the line. Two lines cross at the point that fits both rules.
        

          Check every column. A rule must work for every column of the table, not just the first one.
        

How to graph a rule

Choose some $x$-values and make a table.
Substitute each $x$ into the rule to get $y$.
Plot each column as a point $(x, y)$.
Join the points with a ruler and put arrows on both ends.

How to find the rule from a table

Find the step: how much $y$ changes when $x$ goes up $1$.
Multiply $x$ by the step (use $-$ if $y$ goes down).
Add or take away to match the $y$-value at $x = 0$.
Check the rule on every column.

Example 1 — Table to graph

Complete the table for $y = x + 3$, then plot and join the points.

$x$	$-1$	$0$	$1$	$2$
$y$

Solution

$-1 + 3$	$=$	$2$
$0 + 3$	$=$	$3$
$1 + 3$	$=$	$4$
$2 + 3$	$=$	$5$

- 1 + 3 = 2, 0 + 3 = 3, 1 + 3 = 4, 2 + 3 = 5

The $y$-values are $2, 3, 4, 5$. The points $(-1, 2)$, $(0, 3)$, $(1, 4)$, $(2, 5)$ make a straight line.

Example 2 — Describe the pattern

Graph the table of values and describe the pattern of points.

$x$	$1$	$2$	$3$	$4$	$5$
$y$	$3$	$6$	$9$	$12$	$15$

Solution

The ordered pairs are $A(1, 3)$, $B(2, 6)$, $C(3, 9)$, $D(4, 12)$, $E(5, 15)$.

The points are collinear: they lie on a straight line. Each $y$ is three times $x$, so the rule is $y = 3x$.

Example 3 — A rule that goes down

(i) Find the rule for this table.
(ii) Use the rule to find $y$ when $x = 6$.

$x$	$0$	$1$	$2$	$3$
$y$	$11$	$8$	$5$	$2$

Solution

(i) $y$ goes down $3$ each time $x$ goes up $1$. At $x = 0$, $y = 11$. So $y = 11 - 3x$. Check with $x = 3$: $11 - 3 \times 3 = 2$.

$y$	$=$	$11 - 3 \times 6$
	$=$	$11 - 18 = -7$

y = 11 - 3 \times 6 = 11 - 18 = - 7

(i) $y = 11 - 3x$
(ii) $y = -7$

Example 4 — Horizontal and vertical lines

Describe the line through each set of points.
(i) $x = -1, 0, 1, 2$ and $y = 4, 4, 4, 4$
(ii) $x = -3, -3, -3$ and $y = 0, 1, 2$

Solution

(i) Every $y$ is $4$: a horizontal line.
(ii) Every $x$ is $-3$: a vertical line.

(i) $y = 4$
(ii) $x = -3$

Example 5 — Half steps and extending

The points $(0, 1)$, $(2, 2)$ and $(4, 3)$ are plotted and joined. The line is extended. What is $y$ when $x = 8$?

Solution

Each time $x$ goes up $2$, $y$ goes up $1$. From $x = 4$ to $x = 8$ is two more steps of $2$, so $y$ goes up $2$ more.

$y$

$=$

$3 + 1 + 1 = 5$

y = 3 + 1 + 1 = 5

$y = 5$ when $x = 8$: the point $(8, 5)$ is on the line.

Example 6 — A matchstick pattern

Triangles in a row are made from matches: $1$ triangle uses $3$ matches, $2$ use $5$, $3$ use $7$, $4$ use $9$.
(i) Write a rule for the number of matches $M$ for $t$ triangles.
(ii) How many matches make $20$ triangles?
(iii) How many triangles can $61$ matches make?

Solution

(i) $M$ goes up $2$ each time, so start with $2t$: $2 \times 1 = 2$, and the table has $3$ ($1$ more). So $M = 2t + 1$.
(iii) Undo the rule: take away $1$, then divide by $2$.

$M$	$=$	$2 \times 20 + 1 = 41$
$(61 - 1) \div 2$	$=$	$30$

M = 2 \times 20 + 1 = 41, (61 - 1) \div 2 = 30

(i) $M = 2t + 1$
(ii) $41$ matches
(iii) $30$ triangles

Example 7 — Two phone plans

Plan A costs $\$10$ plus $\$2$ per GB of data: $A = 2g + 10$. Plan B costs $\$4$ per GB: $B = 4g$. For how many GB do they cost the same?

Solution

Make a table and find the column where both costs are equal.

GB $(g)$	$3$	$4$	$5$	$6$
Plan A ($)	$16$	$18$	$\mathbf{20}$	$22$
Plan B ($)	$12$	$16$	$\mathbf{20}$	$24$

$2 \times 5 + 10$	$=$	$20$
$4 \times 5$	$=$	$20$

2 \times 5 + 10 = 20, 4 \times 5 = 20

They cost the same, $\$20$, for $5$ GB. On a graph, the two lines cross at $(5, 20)$.

Example 8 — Is the point on the line?

The rule is $y = 4x - 3$. Is each point on the line?
(i) $(15, 57)$
(ii) $(12, 44)$

Solution

$4 \times 15 - 3$	$=$	$57$
$4 \times 12 - 3$	$=$	$45$

4 \times 15 - 3 = 57, 4 \times 12 - 3 = 45

(i) Yes: $x = 15$ gives $y = 57$.
(ii) No: $x = 12$ gives $y = 45$, not $44$.

Common pitfalls

Swapping x and y. The column $x = 2$, $y = 3$ is the point $(2, 3)$, not $(3, 2)$.

Checking only one column. $y = x$ fits $(1, 1)$, but for $x = 2$ it gives $2$, not $3$. The rule that fits $(1, 1)$, $(2, 3)$, $(3, 5)$ is $y = 2x - 1$.

Mixing up horizontal and vertical. If the $y$-values are $3, 3, 3$, the line is $y = 3$ (horizontal), not $x = 3$.

Joining with a curve or no arrows. Join collinear points with a ruler, and put arrows on both ends because the line keeps going.

Frequently asked questions

What is a table of values?

A table of $x$-values and the $y$-values a rule gives for them. Each column is a point $(x, y)$.

How do you complete a table of values?

Substitute each $x$-value into the rule. For $y = x + 3$ and $x = 2$, $y = 5$.

How do you find the rule from a table?

Find how much $y$ changes when $x$ goes up $1$, multiply $x$ by that step, then add or take away to match the table.

What line do the points (5, 1), (5, 2), (5, 3) make?

The vertical line $x = 5$, because every $x$ is $5$.

How do you check if a point is on a line?

Substitute the $x$-value into the rule. If the answer is the $y$-value, the point is on the line.

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Transformations

\(x\)	\(-1\)	\(0\)	\(2\)	\(4\)
\(y\)	\(0\)	\(3\)	\(5\)	\(1\)

\(x\)	\(0\)	\(1\)	\(2\)	\(3\)
\(y\)	\(-1\)	\(1\)	\(3\)	\(5\)

\(x\)	\(-1\)	\(0\)	\(1\)	\(2\)
\(y\)

\(-1 + 3\)	\(=\)	\(2\)
\(0 + 3\)	\(=\)	\(3\)
\(1 + 3\)	\(=\)	\(4\)
\(2 + 3\)	\(=\)	\(5\)

\(x\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)
\(y\)	\(3\)	\(6\)	\(9\)	\(12\)	\(15\)

\(M\)	\(=\)	\(2 \times 20 + 1 = 41\)
\((61 - 1) \div 2\)	\(=\)	\(30\)

GB \((g)\)	\(3\)	\(4\)	\(5\)	\(6\)
Plan A ($)	\(16\)	\(18\)	\(\mathbf{20}\)	\(22\)
Plan B ($)	\(12\)	\(16\)	\(\mathbf{20}\)	\(24\)

\(4 \times 15 - 3\)	\(=\)	\(57\)
\(4 \times 12 - 3\)	\(=\)	\(45\)

\(y\)	\(=\)	\(11 - 3 \times 6\)
\(\)	\(=\)	\(11 - 18 = -7\)

Graphing Table of Values

📖 Theory

How to graph a rule

How to find the rule from a table

Common pitfalls

Frequently asked questions

Theory