Find the rule for each of the following sets of ordered pairs and tables of values.

Here are the solutions for Question 2 and Question 3.

Question 2

2.1

a) To find the rule, we observe the relationship between $x$ and $y$ from the given pairs: $(-1, -3)$, $(0, -1)$, $(1, 1)$.
The difference in $y$ for a unit increase in $x$ is:
$-1 - (-3) = 2$
$1 - (-1) = 2$
This indicates a slope of 2. So, the rule is in the form $y = 2x + c$.
Using the point $(0, -1)$:

$$-1 = 2(0) + c$$ $$-1 = c$$

Thus, the rule is $y = 2x - 1$.

$$\boxed{y = 2x - 1}$$

b) The rule states that to get the output value ($y$), you multiply the input value ($x$) by 2 and then subtract 1.

c) Using the rule $y = 2x - 1$:
To find $m$: When $y = 3$, $x = m$.

$$3 = 2m - 1$$ $$3 + 1 = 2m$$ $$4 = 2m$$ $$m = 2$$

To find $n$: When $x = 3$, $y = n$.

$$n = 2(3) - 1$$ $$n = 6 - 1$$ $$n = 5$$ $$\boxed{m = 2, n = 5}$$

d) The ordered pairs are:

$$(-1; -3), (0; -1), (1; 1), (2; 3), (3; 5)$$

e) The relationship as a flow diagram:

$$x \xrightarrow{\text{Multiply by 2}} \text{Result} \xrightarrow{\text{Subtract 1}} y$$

2.2
To determine the rule for the table:

x	1	2	3	4	5
y	4	6	8	10	12
The difference in $y$ for a unit increase in $x$ is:
$6 - 4 = 2$
$8 - 6 = 2$
This indicates a slope of 2. So, the rule is in the form $y = 2x + c$.
Using the point $(1, 4)$:

$$4 = 2(1) + c$$ $$4 = 2 + c$$ $$c = 2$$

Thus, the rule is $y = 2x + 2$.

$$\boxed{y = 2x + 2}$$

Question 3

3.1
Complete the table for each equation:

For $y = -4x$:

• If $x = -1$, $y = -4(-1) = 4$
• If $x = 0$, $y = -4(0) = 0$
• If $x = 1$, $y = -4(1) = -4$

For $y = -4x + 1$:

• If $x = -1$, $y = -4(-1) + 1 = 4 + 1 = 5$
• If $x = 0$, $y = -4(0) + 1 = 0 + 1 = 1$
• If $x = 1$, $y = -4(1) + 1 = -4 + 1 = -3$

The completed table:

x	-2	-1	0	1
$y = -4x$	8	4	0	-4
$y = -4x + 1$	9	5	1	-3
Ordered pairs For $y = -4x + 1$	(-2; 9)	(-1; 5)	(0; 1)	(1; -3)