To determine the rate of change for the interval between [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex] for the quadratic equation [tex]\(f(x) = 2x^2 + x - 3\)[/tex], we follow these steps:
1. Identify the function values at the given points:
- At [tex]\(x = 0\)[/tex], using the table, we see that [tex]\(f(0) = -3\)[/tex].
- At [tex]\(x = 2\)[/tex], using the table, we see that [tex]\(f(2) = 7\)[/tex].
2. Use the function values to calculate the rate of change:
The formula for the rate of change (or the average rate of change) over an interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[
\text{Rate of Change} = \frac{f(b) - f(a)}{b - a}
\][/tex]
For our specific case:
[tex]\[
a = 0 \quad \text{and} \quad b = 2
\][/tex]
So, we plug in the function values:
[tex]\[
\text{Rate of Change} = \frac{f(2) - f(0)}{2 - 0}
\][/tex]
Substituting the values we found:
[tex]\[
\text{Rate of Change} = \frac{7 - (-3)}{2 - 0} = \frac{7 + 3}{2} = \frac{10}{2} = 5
\][/tex]
Therefore, the rate of change for the interval between [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex] for the function [tex]\(f(x) = 2x^2 + x - 3\)[/tex] is [tex]\(\boxed{5}\)[/tex].
