8 In this graph what will happen to the value of y when the value of x decrease? 9 State whether the graph is an increase; a decrease or a constant.

The problem describes structures where squares represent families and dots represent material. Let's analyze the pattern for the number of squares (families) and dots (material).

• Structure 1: 1 square (1x1 grid of squares). It has 4 dots, forming a 2x2 grid of dots.
• Structure 2: 2 squares (1x2 grid of squares, i.e., 1 column, 2 rows). It has 6 dots, forming a 2x3 grid of dots.
• Structure 3: 4 squares (2x2 grid of squares). It has 9 dots, forming a 3x3 grid of dots.
• Structure 4: 9 squares (3x3 grid of squares). It has 16 dots, forming a 4x4 grid of dots.

From this, we can observe a consistent pattern for the number of dots:
If a structure is a $w \times h$ grid of squares, the number of dots is $(w+1)(h+1)$.
For square grids (where $w=h=k$), the number of dots is $(k+1)^2$.

For structures 3 and 4, the number of squares is a $k \times k$ grid where $k = n-1$.
So, for $n \ge 3$, the number of squares is $(n-1)^2$, and the number of dots is $((n-1)+1)^2 = n^2$.
Let's verify this:
For $n=3$: $k=2$. Squares = $(3-1)^2 = 4$. Dots = $3^2 = 9$. (Matches)
For $n=4$: $k=3$. Squares = $(4-1)^2 = 9$. Dots = $4^2 = 16$. (Matches)

For structures 1 and 2:
For $n=1$: 1 square (1x1 grid). Here $k=1$. Dots = $(1+1)^2 = 4$. (Matches)
For $n=2$: 2 squares (1x2 grid). Here $w=1, h=2$. Dots = $(1+1)(2+1) = 2 \times 3 = 6$. (Matches)

It appears the "general rule" for the $n^{th}$ structure (for $n \ge 3$) is that it's an $(n-1) \times (n-1)$ grid of squares, and thus has $n^2$ dots. Structure 1 fits the square grid rule. Structure 2 is an exception to the square grid of squares, but fits the general $w \times h$ grid of squares rule. For questions asking for a general rule for the $n^{th}$ structure, it typically refers to the pattern observed in the later, more systematic structures.

The image shows 4 dots.

The image shows 6 dots.

The image shows 9 dots.

The image shows 16 dots.

Based on the pattern for structures 3 and 4, where structure $n$ is an $(n-1) \times (n-1)$ grid of squares:
Structure 5 will be a $(5-1) \times (5-1) = 4 \times 4$ grid of squares.
The dots will form a $5 \times 5$ grid of vertices.

+---+---+---+---+
| . | . | . | . | .
+---+---+---+---+
| . | . | . | . | .
+---+---+---+---+
| . | . | . | . | .
+---+---+---+---+
| . | . | . | . | .
+---+---+---+---+
. . . . .

(Imagine the dots are at the intersections of the lines, forming a 5x5 grid of dots, and the squares are formed by these lines.)

The structures grow in size.

• The number of squares (families) increases: Structure 1 has 1 square, Structure 2 has 2 squares, Structure 3 has 4 squares (a 2x2 grid), and Structure 4 has 9 squares (a 3x3 grid).
• The number of dots (material) also increases: Structure 1 has 4 dots, Structure 2 has 6 dots, Structure 3 has 9 dots, and Structure 4 has 16 dots.
• For structures that are a $k \times k$ grid of squares (S1, S3, S4), the number of dots is $(k+1)^2$. Structure 2 is a $1 \times 2$ grid of squares, and its dots are $(1+1)(2+1) = 6$.

Assuming the pattern for $n \ge 3$ (where the structure is a square grid of squares) is the intended "general rule":

This question requires a subjective choice based on the provided information. The goal is usually to house as many families as possible with reasonable material.
Let's consider the ratio of families to material (squares to dots).

• Structure 1: 1 family / 4 dots = 0.25 families/dot
• Structure 2: 2 families / 6 dots = 0.33 families/dot
• Structure 3: 4 families / 9 dots = 0.44 families/dot
• Structure 4: 9 families / 16 dots = 0.56 families/dot
• Structure 5: 16 families / 25 dots = 0.64 families/dot
• Structure 6: 25 families / 36 dots = 0.69 families/dot
• Structure $n$: $(n-1)^2$ families / $n^2$ dots. Ratio = $\frac{(n-1)^2}{n^2} = \left(\frac{n-1}{n}\right)^2 = \left(1 - \frac{1}{n}\right)^2$.

As $n$ increases, the ratio of families to dots approaches 1, meaning it becomes more efficient. Therefore, a larger structure is more efficient in terms of material per family.

I would choose Structure 6 (or any larger structure if available).
Reason: Structure 6 houses 25 families using 36 dots of material. This provides a better ratio of families housed per unit of material compared to smaller structures. As the structure number increases, the efficiency of housing families per dot of material also increases, making larger structures more cost-effective in terms of material.

3.1. Triangle ABC on the Cartesian plane below represents a roof for one unit. To extend the roof to cater for 16 unit you need to enlarge the roof using $(y+3; x+3)$. Calculate the coordinates of the enlarged roof.

The original coordinates of Triangle ABC are not provided in the image. I need the coordinates of A, B, and C to apply the transformation. Please provide the coordinates of Triangle ABC.