To find the rate of change between the given sets of input and output values, let's examine the data carefully.
We are given the following data:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Input} & \text{Output} \\
\hline
1 & 11 \\
\hline
2 & 13 \\
\hline
3 & 15 \\
\hline
4 & 17 \\
\hline
5 & 19 \\
\hline
\end{array}
\][/tex]
We can see that as the input increases, the output also increases. To find the rate of change, we can use the concept of the slope in a straight line, which is given by:
[tex]\[
\text{Rate of Change} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Here, [tex]\(y_1\)[/tex] and [tex]\(y_2\)[/tex] are the output values corresponding to [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex] which are the input values.
Let's choose the first and the last data points to compute the rate of change:
- [tex]\((x_1, y_1) = (1, 11)\)[/tex]
- [tex]\((x_2, y_2) = (5, 19)\)[/tex]
Plug these values into the formula:
[tex]\[
\text{Rate of Change} = \frac{19 - 11}{5 - 1}
\][/tex]
Now, calculate the differences in the numerator and denominator:
[tex]\[
\Delta y = 19 - 11 = 8
\][/tex]
[tex]\[
\Delta x = 5 - 1 = 4
\][/tex]
Thus, the rate of change is:
[tex]\[
\text{Rate of Change} = \frac{8}{4} = 2
\][/tex]
Therefore, the rate of change for the given data is [tex]\(2.0\)[/tex]. This means that for every unit increase in the input, the output increases by 2 units.
