1. Section 6.1
Exercise 32 Page 235
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The pattern of growth tells us how many more tiles the next figure in a tile pattern has.
If point (x,y) lies on the given graph, then Figure x has y tiles.
For a tile pattern with a rule in the form y = mx + b, the value of m is the pattern of growth. Then, b is the y-intercept. That tells us the number of tiles Figure 0 has.
The pattern of growth tells us how many more tiles the next figure in a tile pattern has.
Pattern of Growth: 5
Figure : 3 tiles
Pattern of Growth: -2
Figure : 3 tiles
Pattern of Growth: 3
Figure : -14 tiles
Pattern of Growth: -5
Figure : 3 tiles
Let's take a look at the given diagram.
The diagram reveals a tile pattern. Each succeeding figure — Figure 2 to 3 and so on — has two tiles added to the top, two more tiles added to the bottom, and one additional tile tacked on to the right. Wait, there is more to discover!
Each new figure has 5 more tiles than the one that came before. This means that the pattern of growth is 5. We are not finished. We are also looking for the number of tiles of Figure 0. Let's first find the number of tiles of Figure 2.
Figure 2 has 13 tiles. We know that the pattern of growth is 5. That means Figure 1 has 5 tiles less than Figure 2. Figure 1: 13 - 5 = 8 tiles Similarly, Figure 0 has 5 tiles less than Figure 1. Figure 0: 8 - 5 = 3 tiles
The following graph describes a tile pattern.
If point (x,y) lies on such a graph, it tells us that Figure x has y tiles. We want to find the pattern of growth of this tile pattern and the number of tiles Figure 0 has. The number of tiles Figure 0 has is the y-intercept, so let's find where our graph intercepts the y-axis.
Our graph intercepts the y-axis at point ( 0, 3), so 3 is the number of tiles Figure 0 has. Here, the pattern of growth tells us how the y-values change if in we increase the x-value by one. We know the number of tiles of Figure 0 , so let's find the number of tiles Figure 1 has. We will find at which point the graph intersect the line x=1.
The graph intersect the line x=1 at point ( 1, 1). Therefore, Figure 1 has 1 tile. To find the pattern of growth, we should subtract from this value the number of tiles Figure 0 has, which is 3. Growth Rate: 1 - 3 = -2
The following rule describes a tile pattern. y=3x-14 If a rule has a form y = mx + b, the value of m is the pattern of growth, and b is the number of tiles Figure 0 has. Let's find these values in our rule. y=3x-14 ⇔ y = 3x + ( -14) In our case, m= 3 and b= -14.
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y | 18 | 13 | 8 | 3 | -2 | -7 | -12 |
We want to find the pattern of growth and the number of tiles Figure 0 has. Notice that for x = 0, we have y = 3. This means that Figure 0 has 3 tiles. Figure : 3tiles Next, for x=1, we have y = -2. That means Figure 1 has -2 tiles. To find the pattern of growth, we should subtract the number of tiles in Figure 1 from the number of tiles in Figure 0. Let's do it! Growth Rate: -2 - 3 = -5