Question

Which type of function best models the data in the table?

Type of Function

x y
2
−58.4

4
−82.8

6
−123.2

8
−179.6

(1 point)
Responses

linear
linear

quadratic
quadratic

exponential
exponential

cubic

Answer 1

To determine which type of function best models the given data, we need to first check the differences between the y-values. This will help us identify whether the relationship is linear, quadratic, exponential, or cubic.

1. **Linear** functions have a constant first difference (the differences between consecutive y-values remain the same).

2. **Quadratic** functions have a constant second difference (the differences of the first differences are constant).

3. **Exponential** functions do not have constant differences, but rather constant ratios between consecutive y-values.

4. **Cubic** functions have constant third differences.

Let's calculate the differences for the given data:

**Given data:**
- For \(x = 2\), \(y = -58.4\)
- For \(x = 4\), \(y = -82.8\)
- For \(x = 6\), \(y = -123.2\)
- For \(x = 8\), \(y = -179.6\)

**First Differences:**
- From \(x=2\) to \(x=4\): \(-82.8 - (-58.4) = -24.4\)
- From \(x=4\) to \(x=6\): \(-123.2 - (-82.8) = -40.4\)
- From \(x=6\) to \(x=8\): \(-179.6 - (-123.2) = -56.4\)

First differences:
- \(-24.4\)
- \(-40.4\)
- \(-56.4\)

**Second Differences:**
- From \(-24.4\) to \(-40.4\): \(-40.4 - (-24.4) = -16\)
- From \(-40.4\) to \(-56.4\): \(-56.4 - (-40.4) = -16\)

Second differences:
- \(-16\)
- \(-16\)

Since the second differences are constant, the data can be modeled by a **quadratic function**.

Therefore, the correct response is:

**quadratic**.