Sure! To determine the rate of change of a function over the interval between 2 and 6 on the [tex]\( x \)[/tex]-axis, consider the following provided specific choices:
- [tex]\( -3 \)[/tex]
- [tex]\( -\frac{1}{3} \)[/tex]
- [tex]\( \frac{1}{3} \)[/tex]
- [tex]\( 3 \)[/tex]
Each of these represents potential rates of change, which are essentially the slopes of the line connecting two points on a graph over the specified interval.
The rate of change is calculated by the formula:
[tex]\[ \text{Rate of Change} = \frac{\Delta y}{\Delta x} \][/tex]
Where:
- [tex]\(\Delta y\)[/tex] represents the change in the [tex]\( y \)[/tex]-values
- [tex]\(\Delta x\)[/tex] represents the change in the [tex]\( x \)[/tex]-values
Given that we are interested in the rate of change over the interval between [tex]\( x = 2 \)[/tex] and [tex]\( x = 6 \)[/tex], let's consider the choices again:
The options are: [tex]\( -3 \)[/tex], [tex]\( -\frac{1}{3} \)[/tex], [tex]\( \frac{1}{3} \)[/tex], and [tex]\( 3 \)[/tex].
From the given choices, the rate of change values are:
- [tex]\( -3 \)[/tex]
- [tex]\( -0.333 \)[/tex] (which corresponds to [tex]\( -\frac{1}{3} \)[/tex])
- [tex]\( 0.333 \)[/tex] (which corresponds to [tex]\( \frac{1}{3} \)[/tex])
- [tex]\( 3 \)[/tex]
So possible answers for the rate of change between [tex]\( x = 2 \)[/tex] and [tex]\( x = 6 \)[/tex] are these values.
Therefore, in the context provided, the rate of change could be found among the given choices:
[tex]\[ -3, -\frac{1}{3}, \frac{1}{3}, \text{ or } 3 \][/tex].
Thus, the rate of change for the interval between 2 and 6 on the [tex]\( x \)[/tex]-axis is [tex]\( -3, -\frac{1}{3}, \frac{1}{3}, \)[/tex] or [tex]\( 3 \)[/tex].
