Compare the linear function and the arithmetic sequence below.
Linear Function
x y
-1 7
0 9
1 11
2 13
The rate of change of the linear function is
The y-intercept of the linear function is
Arithmetic Sequence:
an = 9 + (n−1)(2)
The rate of change (common difference) of the arithmetic sequence is
The initial value of the arithmetic sequence is
The rate of change of this linear function is
the rate of change of this arithmetic sequence.
Arithmetic sequences
can be represented as a linear function.
The rate of change of the linear function is the difference in the y-values over the difference in the x-values. In this case, the difference in y-values is 2 and the difference in x-values is 1. Therefore, the rate of change is 2/1 = 2.
The y-intercept of the linear function is the value of y when x is 0. In this case, when x is 0, y is 9. Therefore, the y-intercept is 9.
The rate of change (common difference) of the arithmetic sequence is the difference between consecutive terms. In this case, the common difference is 2.
The initial value of the arithmetic sequence is the value when n is 1. Plugging in n=1 into the sequence formula, we get a1 = 9 + (1-1)(2) = 9. Therefore, the initial value is 9.
The rate of change of this linear function is the same as the rate of change (common difference) of the arithmetic sequence. In this case, both are 2.
To compare the linear function and the arithmetic sequence, let's look at their respective characteristics.
Linear Function:
The given linear function can be represented in the form of y = mx + b, where m represents the rate of change (or slope) and b represents the y-intercept.
To determine the rate of change of the linear function, we can choose any two data points from the given table and calculate the change in y divided by the change in x. Let's consider the points (-1, 7) and (0, 9). The change in y is 2, and the change in x is 1. Therefore, the rate of change (or slope) is 2.
To find the y-intercept, we can notice that when x = 0, y = 9. Hence, the y-intercept is 9.
Therefore, the rate of change of the linear function is 2, and the y-intercept is 9.
Arithmetic Sequence:
The given arithmetic sequence is represented by the formula an = 9 + (n-1)(2), where n is the number of terms.
To find the rate of change (common difference) of the arithmetic sequence, we can observe that the difference between consecutive terms is 2.
The initial value of the arithmetic sequence is given by the first term, a1, which can be calculated by substituting n = 1 in the formula. Therefore, a1 = 9 + (1-1)(2) = 9.
Comparing Rate of Change:
Since the linear function has a constant rate of change of 2, while the arithmetic sequence also has a constant difference of 2 between consecutive terms, we can conclude that the rate of change of this linear function is equal to the rate of change of this arithmetic sequence.
Hence, the rate of change of this linear function is the same as the rate of change of this arithmetic sequence.
Moreover, it is important to recognize that arithmetic sequences can be represented as linear functions. In this case, the formula for the arithmetic sequence can be transformed into a linear function.