Question

Compare the linear function and the arithmetic sequence below.

Linear Function:

x y
-1 10
0 14
1 18
2 22

The rate of change of the linear function is

The y-intercept of the linear function is

Arithmetic Sequence (Explicit formula):

an = 10+ 4(n−1)

The rate of change (common difference) of the arithmetic sequence is

The first value of the arithmetic sequence is
The rate of change of this linear function is BLANK the rate of change of this arithmetic sequence.

Arithmetic sequences BLANK can be represented as a linear function.

Answer 1

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

### For the Linear Function:

We have the data points:

| x | y |
|----|----|
| -1 | 10 |
| 0 | 14 |
| 1 | 18 |
| 2 | 22 |

1. **Rate of Change of the Linear Function:**
- The rate of change (slope) can be calculated using any two points.
- Using points (0, 14) and (1, 18):
\[
\text{Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{18 - 14}{1 - 0} = 4
\]

2. **Y-Intercept of the Linear Function:**
- The y-intercept is the value of y when x = 0.
- From the table, when \( x = 0 \), \( y = 14 \).
- Thus, the y-intercept is 14.

### For the Arithmetic Sequence:

The explicit formula is:
\[
a_n = 10 + 4(n - 1)
\]

1. **Rate of Change (Common Difference) of the Arithmetic Sequence:**
- The common difference is the coefficient of \( n \) in the explicit formula.
- Hence, the rate of change (common difference) is 4.

2. **First Value of the Arithmetic Sequence:**
- The first term (when \( n = 1 \)) can be calculated as follows:
\[
a_1 = 10 + 4(1 - 1) = 10 + 0 = 10
\]
- Thus, the first value is 10.

### Comparison of Rates of Change:

- The rate of change of the linear function is **4**.
- The rate of change of the arithmetic sequence is **4**.

**Conclusion:**
- The statement "The rate of change of this linear function is _the rate of change of this arithmetic sequence."** can be filled with "equal to".
- Additionally, "Arithmetic sequences _can be represented as a linear function" can be filled with "can", as arithmetic sequences indeed can be represented as linear functions.

### Final Summary:

- The rate of change of the linear function is **4**.
- The y-intercept of the linear function is **14**.
- The rate of change (common difference) of the arithmetic sequence is **4**.
- The first value of the arithmetic sequence is **10**.
- The rate of change of this linear function is **equal to** the rate of change of this arithmetic sequence.
- Arithmetic sequences **can** be represented as a linear function.