Question

x y
2 7
3 10.5
4 14
5 17.5
6 21


Which function has a greater rate of change than the function represented in the table.
Responses
A y = 4xy = 4x
B y = x + 2y = x + 2
C y = 2x + 3y = 2x + 3
D y = 3x + 4

Answer 1

To determine which function has a greater rate of change than the function represented in the table, we first need to find the rate of change in the table.

The values of \(y\) in the table corresponding to \(x\) values are:

- When \(x = 2\), \(y = 7\)
- When \(x = 3\), \(y = 10.5\)
- When \(x = 4\), \(y = 14\)
- When \(x = 5\), \(y = 17.5\)
- When \(x = 6\), \(y = 21\)

The change in \(y\) is consistent as \(x\) increases. Let's calculate the rate of change between two consecutive points as an example:

From \(x = 2\) to \(x = 3\):
\[
\text{Change in } y = 10.5 - 7 = 3.5
\]
\[
\text{Change in } x = 3 - 2 = 1
\]
Thus, the rate of change from \(x = 2\) to \(x = 3\) is:
\[
\text{Rate of change} = \frac{3.5}{1} = 3.5
\]

Repeating this for other intervals shows the rate of change remains \(3.5\) across all intervals.

Now we will evaluate the rate of change of the provided functions:

**A.** \(y = 4x\)
- The rate of change (slope) is \(4\).

**B.** \(y = x + 2\)
- The rate of change (slope) is \(1\).

**C.** \(y = 2x + 3\)
- The rate of change (slope) is \(2\).

**D.** \(y = 3x + 4\)
- The rate of change (slope) is \(3\).

Now we compare the rates of change:
- Function A: \(4\) (greater than \(3.5\))
- Function B: \(1\) (less than \(3.5\))
- Function C: \(2\) (less than \(3.5\))
- Function D: \(3\) (less than \(3.5\))

Thus, the only function with a rate of change greater than \(3.5\) is:

**Answer: A. \(y = 4x\)**.