Question

4-38. look at the graph at right. what statements can you make about the tile pattern the graph represents? how many tiles are in figure 0? figure 1? what is the pattern of growth? what is the rule for the pattern? use your pattern skills to complete the table below.

Explanation:

Step1: Analyze Figure 0

From the graph, when Figure Number is 0, the Number of Tiles is 7? Wait, no, looking at the graph, the first point (Figure 0) seems to be at (0,7)? Wait, no, the x - axis is Figure Number (vertical) and y - axis is Number of Tiles (horizontal)? Wait, the graph has Figure Number on the vertical (y - axis? Wait, no, the labels: "Figure Number" is vertical, "Number of Tiles" is horizontal. Wait, the first point: when Figure Number is 0, Number of Tiles is 7? Wait, no, the grid: let's check the coordinates. Let's assume Figure Number (n) is on the y - axis (vertical) and Number of Tiles (t) is on the x - axis (horizontal). Wait, the first plotted point: when n = 0, t = 7? Wait, no, the first visible point: when n = 0, t = 7? Wait, the graph shows a line. Let's find the slope. Let's take two points. Let's say when n = 0, t = 7; when n = 1, t = 9? Wait, no, the graph: let's look at the points. Wait, the first point (Figure 0) is at (7, 0)? No, maybe I got the axes reversed. Let's re - orient: usually, the independent variable (Figure Number) is on the x - axis, and dependent (Number of Tiles) on the y - axis. Wait, the label "Figure Number" is vertical (so y - axis) and "Number of Tiles" is horizontal (x - axis). So the coordinates are (t, n), where t is Number of Tiles (x - axis) and n is Figure Number (y - axis). So the first point: t = 7, n = 0 (Figure 0, 7 tiles). Then when n = 1, t = 9? Wait, no, the line is increasing? Wait, the graph shows a line with positive slope. Wait, let's check the points. Let's take two points: when n = 0, t = 7; when n = 1, t = 9? Wait, no, the grid: each square is 1 unit. Let's see, the first point (Figure 0) is at (7, 0) (x = 7, y = 0), then next point (x = 9, y = 1), (x = 11, y = 2), etc. Wait, so the pattern: for Figure 0 (n = 0), Number of Tiles (t)=7? Wait, no, maybe the axes are reversed. Let's assume Figure Number (n) is on the x - axis, Number of Tiles (t) on the y - axis. Then the first point: n = 0, t = 7; n = 1, t = 9; n = 2, t = 11, etc. So the slope (rate of change) is (9 - 7)/(1 - 0)=2. So the number of tiles increases by 2 as the figure number increases by 1.

Step2: Find tiles in Figure 0 and Figure 1

From the graph, when Figure Number (n)=0, the Number of Tiles (t) is 7? Wait, no, looking at the graph, the first point (when Figure Number is 0) is at 7? Wait, the x - axis (Number of Tiles) has marks 2,4,6,8,10,12,14,16. The y - axis (Figure Number) has 0,1,2,3,4,5,6. The first plotted point is at (7, 0)? No, maybe the first point is at (7, 0) (Number of Tiles = 7, Figure 0), then when Figure Number = 1, Number of Tiles = 9? Wait, the line is going up with a slope of 2. So Figure 0: Let's check the graph again. The first point (lowest Figure Number, 0) is at Number of Tiles = 7? Wait, maybe the initial point (Figure 0) has 7 tiles, and Figure 1 has 9 tiles? Wait, no, maybe I made a mistake. Wait, the problem says "How many tiles are in Figure 0? Figure 1?". Let's look at the graph: the first point (when Figure Number is 0) is at Number of Tiles = 7? Wait, the grid: the first point is at (7, 0) (x = 7, y = 0), so Figure 0 has 7 tiles? Then Figure 1: moving up 1 in Figure Number, the Number of Tiles increases by 2 (since the slope is 2). So Figure 1: 7+2 = 9? Wait, no, maybe the axes are reversed. Let's assume Figure Number is on the x - axis, Number of Tiles on the y - axis. Then when x = 0 (Figure 0), y = 7; x = 1 (Figure 1), y = 9. So Figure 0: 7 tiles, Figure 1: 9 tiles.

Step3: Pattern of growth

The number of tiles increases by 2 as the figure number increases by 1. So it's a linear growth with a common difference of 2.

Step4: Rule for the pattern

Let t be the number of tiles, n be the figure number. The general form of a linear equation is t = mn + b, where m is the slope and b is the y - intercept. We found m = 2 (since for each increase in n by 1, t increases by 2). When n = 0, t = 7 (from Figure 0), so b = 7. So the rule is t = 2n+7.

Answer:

• Figure 0: 7 tiles
• Figure 1: 9 tiles
• Pattern of growth: The number of tiles increases by 2 for each increase in figure number by 1 (linear growth with common difference 2).
• Rule: Let \( n \) be the figure number and \( t \) be the number of tiles. The rule is \( t = 2n + 7 \).