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Question

1. A graph of the sequence f(n)=2n - 3 and a graph of the function f(x)=2x - 3 are both shown below.

1. a. What similarity or similarities exist between the two graphs?
2. b. What difference(s) exist between the two graphs?
3. c. What is the domain and range for each graph?

2. Note the pattern shown below.

1. a. What would the next image look like? Insert it below.
2. b. How many squares would be in the 10th image?
3. c. How many squares would be in the nth image?
4. d. What is the rule for this pattern in terms of n?
5. e. Is the rule arithmetic, geometric, or neither?

Answer 1

1. The two graphs

a. There are strong similarities between the two graphs because the graph on the left corresponds to the graph on the right but discretely sampled, and considering only the values for n >= 0 (interestingly, by mistake on the x-axis even on the left graph it's written x instead of n...). So, both graphs have the same slope and shifted down of 3, but the graph on the left is discrete, while the graph on the right is continuous.

b. The differences are in the fact that the graph on the left is discrete, it is computed for the integer values of n>=0, while the graph on the right is continuous and computed for all the real values of x.

c. For the graph on the left, the domain is all the integer values n>=0, while the range is all the integer values f(n)>=-3. For the graph on the right, both the domain and the range correspond to all the real values ℜ.

1. The pattern of the rectangles

a. It can be observed that the case 1 is a rectangle having 3 squares on the base, and 2 on the height, so we can call it a rectangle (3x2). Case 2 is a rectangle (4x3), and case 3 is a rectangle (5x4). So, every new case has 1 more square on the base, and 1 more square on the height respect to the previous case. Therefore, case 4 will be a rectangle with 6 squares on the base, and 5 squares on the height.

b. We can observe that, given the generic case n the corresponding rectangle has size ((n+2)x(n+1)). In fact, for n=1, case 1 has size ((1+2)x(1+1))=(3x2). Therefore, the 10th image will be ((10+2)x(10+1)) will be a rectangle having size (12x11).

c. We have basically already answered this question before. The nth image will have size ((n+2)x(n+1)). We can find the number of squares by multiplying the size of the base by the size of the height. Squares = (n+2)*(n+1) = n²+3n+2.

d. The rule for this pattern about the size of the base b and of the height h will be respect to n as

b_n = n+2

h_n = n+1

while the rule for the number of squares

Squares_n = n²+3n+2

e. The rule of the size is arithmetic because the difference between two adjacent cases is a constant number, and it's 1

b_n+1 - b_n =1

h_n+1 - h_n =1

while the rule of the number of the squares is neither arithmetic nor geometric because

Squares_n+1 - Squares_n = 2n+4