Algebra A Unit 6 Sample Work Due 10/19/23
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Question
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.
Solution:
Let's start by finding the rate of change and the y-intercept of the linear function.
Rate of Change:
To find the rate of change of a linear function, we can use the formula:
rate of change = (change in y) / (change in x)
Using the given points (-1, 7) and (0, 9), we can find the change in y and change in x as follows:
change in y = 9 - 7 = 2
change in x = 0 - (-1) = 1
Substituting these values into the formula, we get:
rate of change = 2 / 1 = 2
Therefore, the rate of change of the linear function is 2.
Y-intercept:
The y-intercept of a linear function is the value of y when x is 0.
From the given points, we can see that when x is 0, y is 9.
Therefore, the y-intercept of the linear function is 9.
Now let's look at the arithmetic sequence.
Rate of Change (Common Difference):
The common difference in an arithmetic sequence is the difference between any two consecutive terms.
In this case, the common difference is 2.
Initial Value:
The initial value of an arithmetic sequence is the value of the sequence when n is 1.
Using the formula given, we can substitute n = 1 and calculate the initial value:
a1 = 9 + (1 - 1)(2) = 9
Therefore, the initial value of the arithmetic sequence is 9.
Comparing Rate of Change:
The rate of change of the linear function is the same as the common difference in the arithmetic sequence, which is 2.
To summarize:
Linear Function:
- Rate of change: 2
- Y-intercept: 9
Arithmetic Sequence:
- Rate of change (common difference): 2
- Initial value: 9