Question

Compare the linear function and the arithmetic sequence below.
Linear Function:
x | y
-1 | 10
0 | 14
1 | 18
2 | 22
a) What is the rate of change of the linear function?
b) What is the y-intercept of the linear function?
Arthimetic sequence (Explicit formula):
an = 14 + 4 (n - 1)
a) What is the rate of change (common difference) of the arithmetic sequence?
b) What is the initial value of the arithmetic sequence?
c) The rate of change of this linear function is BLANK the rate of change of this arithmetic sequence. What is the BLANK?
d) Arithmetic sequences BLANK can be represented as a linear function. What is the BLANK?

Answer 1

Let's analyze both the linear function and the arithmetic sequence step by step.

### Linear Function:
Given the linear function data:

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

**a) Rate of change of the linear function:**

To find the rate of change (slope) of the linear function, calculate the change in \( y \) divided by the change in \( x \):

Using two points for calculation, for example, (0, 14) and (1, 18):

\[
\text{Rate of change} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{18 - 14}{1 - 0} = \frac{4}{1} = 4
\]

**b) Y-intercept of the linear function:**

The y-intercept is the point where \( x = 0 \). From the table, when \( x = 0 \), \( y = 14 \). Therefore, the y-intercept is:

\[
\text{Y-intercept} = 14
\]

---

### Arithmetic Sequence:
Given the explicit formula for the arithmetic sequence:

\[
a_n = 14 + 4(n - 1)
\]

**a) Rate of change (common difference) of the arithmetic sequence:**

The common difference is the coefficient of \( n \) in the explicit formula. In this case, it is \( 4 \).

**b) Initial value of the arithmetic sequence:**

To find the initial value, substitute \( n = 1 \) into the formula:

\[
a_1 = 14 + 4(1 - 1) = 14 + 4 \cdot 0 = 14
\]

So, the initial value is:

\[
\text{Initial value} = 14
\]

**c) The rate of change of this linear function is BLANK the rate of change of this arithmetic sequence. What is the BLANK?**

We found that the rate of change of the linear function is \( 4 \) and the rate of change (common difference) of the arithmetic sequence is also \( 4 \). Thus, they are equal.

The completed statement is:

\[
\text{The rate of change of this linear function is \textbf{equal to} the rate of change of this arithmetic sequence.}
\]

**d) Arithmetic sequences BLANK can be represented as a linear function. What is the BLANK?**

The correct expression to fill in would be "always".

So, the completed statement is:

\[
\text{Arithmetic sequences \textbf{always} can be represented as a linear function.}
\]

### Summary:
- Rate of change of linear function: **4**
- Y-intercept of linear function: **14**
- Rate of change of arithmetic sequence: **4**
- Initial value of arithmetic sequence: **14**
- The rate of change of this linear function is **equal to** the rate of change of this arithmetic sequence.
- Arithmetic sequences **always** can be represented as a linear function.