Question

A chart shows two points, one at (2, 12) and another at (1, 9). Chart 2 is on this table:

Linear Function 2:

x y

0 2
5 42
Compare Linear Function 1 with Linear Function 2. Which one has the greater rate of change?

(1 point)
Responses

Linear Function 2 has the greater rate of change, with a rate of change equal to 8 while Linear Function 1 has a rate of change equal to 3.
Linear Function 2 has the greater rate of change, with a rate of change equal to 8 while Linear Function 1 has a rate of change equal to 3.

Linear Function 1 has the greater rate of change, with a rate of change equal to 3 while Linear Function 2 has a rate of change equal to 2.
Linear Function 1 has the greater rate of change, with a rate of change equal to 3 while Linear Function 2 has a rate of change equal to 2.

Linear Function 2 has the greater rate of change, with a rate of change equal to 8 while Linear Function 1 has a rate of change equal to 6.
Linear Function 2 has the greater rate of change, with a rate of change equal to 8 while Linear Function 1 has a rate of change equal to 6.

Linear Function 1 has the greater rate of change, with a rate of change equal to 6 while Linear Function 2 has a rate of change equal to 2.

Answer 1

To determine which linear function has a greater rate of change, we need to calculate the rate of change (slope) for both functions.

### Linear Function 1
The two points provided are (2, 12) and (1, 9).

The formula for the rate of change (slope) between two points (x₁, y₁) and (x₂, y₂) is given by:

\[
\text{Rate of Change} = \frac{y₂ - y₁}{x₂ - x₁}
\]

Using the points:
- \( (x₁, y₁) = (1, 9) \)
- \( (x₂, y₂) = (2, 12) \)

Substituting in:

\[
\text{Rate of Change} = \frac{12 - 9}{2 - 1} = \frac{3}{1} = 3
\]

So, the rate of change for Linear Function 1 is **3**.

### Linear Function 2
The two points given for Linear Function 2 are (0, 2) and (5, 42).

Using the same formula:

\[
\text{Rate of Change} = \frac{y₂ - y₁}{x₂ - x₁}
\]

Using the points:
- \( (x₁, y₁) = (0, 2) \)
- \( (x₂, y₂) = (5, 42) \)

Substituting in:

\[
\text{Rate of Change} = \frac{42 - 2}{5 - 0} = \frac{40}{5} = 8
\]

So, the rate of change for Linear Function 2 is **8**.

### Comparison
- **Linear Function 1** has a rate of change of **3**.
- **Linear Function 2** has a rate of change of **8**.

Since 8 is greater than 3, the conclusion is:

**Linear Function 2 has the greater rate of change, with a rate of change equal to 8 while Linear Function 1 has a rate of change equal to 3.**