Question

9. The diagram below shows the steps of tessellation called Sierpinski’s Triangle. The first figure is made up of 1 triangle.a. Write the explicit function that represents the number of small blue triangles the compose figure nb. How many triangles will there be in the 14th figure?c. Consider the function f(x)=3^x. What transformation(s) was/were performed on f(x) to produce your explicit function?

Answer 1

a. We have the following from the diagram

Figure 1 : 1 blue triangle

Figure 2 : 3 blue triangles

Figure 3: 9 blue triangles

Figure 4: 27 blue triangles

We can derive an explicit function for the sequence to be:

`f(n)\text{ = }(3^n)/(3)\text{ }`

b. The number of triangles in the 14th figure

Here we substitute 14 for n into the explicit function

`\begin{gathered} f(n=14)\text{ = }(3^(14))/(3) \\ =\text{ 1594323} \end{gathered}`

There would be 1 594 323 triangles in the 14th figure

c. Given that:

`f(x)=3^x`

Our explicit function:

`f(x)\text{ = }(3^x)/(3)`

The explicit function is multiplied by a factor of 1/3.

Hence, it has been compressed vertically by a factor of 1/3