Question

8. Compare the linear function and the arithmetic sequence below.
Linear Function:
X: -1, 0, 1, 2
Y: 10, 14, 18, 22

The rate of change of the linear function is _____
A) -1
B) 10
C) 4
D) 14
E) 0
The y-intercept of the linear function is _____
A) 0
B) 10
C) -1
D) 14
E) 4

Arithmetic Sequence (Explicit formula): an = 10 + 4(n − 1)
The rate of change (common difference) of the arithmetic sequence is _____
A) 4
B) n
C) n - 1
D) 14
The first term of the arithmetic sequence is _____
A) 10
B) n
C) n - 1
D) 4
The rate of change of this linear function is __________ the rate of change of this arithmetic sequence.
A) equal to
B) greater than
C) less than
Arithmetic sequences __________ can be represented as a linear function.
A) always
B) sometimes
C) never

Answer 1

Let's analyze the linear function and the arithmetic sequence based on the information given.

**Linear Function:**

From the given data:
- X: -1, 0, 1, 2
- Y: 10, 14, 18, 22

1. **Rate of Change**:
To find the rate of change (slope) of the linear function, we can calculate the difference in Y values divided by the difference in X values.

For example, from (0, 14) to (1, 18):
\[
\text{slope} = \frac{(18 - 14)}{(1 - 0)} = \frac{4}{1} = 4
\]
Therefore, the rate of change of the linear function is **4**.

The answer is C) 4.

2. **Y-Intercept**:
The Y-intercept occurs when X = 0. Looking at the given points, when X = 0, Y = 14.

The answer is D) 14.

**Arithmetic Sequence**:

The explicit formula for the arithmetic sequence is given as:
\[
a_n = 10 + 4(n - 1)
\]
This can be simplified to:
\[
a_n = 10 + 4n - 4 = 4n + 6
\]
1. **Rate of Change**:
The rate of change (common difference) of the arithmetic sequence is 4.

The answer is A) 4.

2. **First Term**:
To find the first term of the arithmetic sequence when n = 1, we plug in n = 1 into the formula:
\[
a_1 = 10 + 4(1 - 1) = 10 + 0 = 10
\]
Therefore, the first term of the arithmetic sequence is **10**.

The answer is A) 10.

**Comparison of Rates of Change**:
- The rate of change of the linear function is 4 and the rate of change of the arithmetic sequence is also 4.
- Therefore, the rate of change of the linear function is **equal to** the rate of change of the arithmetic sequence.

The answer is A) equal to.

**Representation of Arithmetic Sequences as Linear Functions**:
Arithmetic sequences are indeed linear functions. Every arithmetic sequence can be represented by a linear function with a constant rate of change.

The answer is A) always.

### Summary of Answers:
- The rate of change of the linear function is **C) 4**.
- The y-intercept of the linear function is **D) 14**.
- The rate of change (common difference) of the arithmetic sequence is **A) 4**.
- The first term of the arithmetic sequence is **A) 10**.
- The rate of change of this linear function is **A) equal to** the rate of change of this arithmetic sequence.
- Arithmetic sequences **A) always** can be represented as a linear function.