To determine which function has a greater rate of change, we need to compare the slopes (rates of change) of Function A and Function B.
Step 1: Determine the rate of change for Function A
The equation of Function A is given by:
[tex]\[ y = 15x + 20 \][/tex]
In this linear equation, the coefficient of [tex]\( x \)[/tex] represents the slope. Therefore, the rate of change (slope) of Function A is:
[tex]\[ \text{Rate of Change of Function A} = 15 \][/tex]
Step 2: Determine the rate of change for Function B
Function B is represented by the table of values:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline$x$ & 0 & 2 & 4 \\
\hline$y$ & 5 & 30 & 55 \\
\hline
\end{tabular}
\][/tex]
We will use the formula for the slope between two points, which is:
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Calculate the slope between points (0, 5) and (2, 30):
[tex]\[ \text{Slope} = \frac{30 - 5}{2 - 0} = \frac{25}{2} = 12.5 \][/tex]
Calculate the slope between points (2, 30) and (4, 55):
[tex]\[ \text{Slope} = \frac{55 - 30}{4 - 2} = \frac{25}{2} = 12.5 \][/tex]
Since the slopes between the different pairs of points are consistent and equal, we can confirm that the rate of change for Function B is:
[tex]\[ \text{Rate of Change of Function B} = 12.5 \][/tex]
Step 3: Compare the rates of change
We have:
- Rate of Change of Function A = 15
- Rate of Change of Function B = 12.5
Comparing these values, we see that:
[tex]\[ 15 > 12.5 \][/tex]
Therefore, Function A has a greater rate of change than Function B.
