Answer :
To compare the rates of change of the two functions, we need to determine the rate of change (slope) for each function step by step.
### Function \(A\)
Function \(A\) is represented by the equation \(y = 2x + 1\).
In this equation, the rate of change (slope) is given by the coefficient of \(x\). Therefore, the rate of change of function \(A\) is:
[tex]\[ \text{Rate of Change of Function } A = 2 \][/tex]
### Function \(B\)
Function \(B\) goes through the points \((1, 3)\), \((3, 11)\), \((4, 15)\), and \((6, 23)\).
To determine the rate of change for function \(B\), we calculate the slope between each pair of consecutive points:
1. Between \((1, 3)\) and \((3, 11)\):
[tex]\[ \text{Slope}_1 = \frac{11 - 3}{3 - 1} = \frac{8}{2} = 4 \][/tex]
2. Between \((3, 11)\) and \((4, 15)\):
[tex]\[ \text{Slope}_2 = \frac{15 - 11}{4 - 3} = \frac{4}{1} = 4 \][/tex]
3. Between \((4, 15)\) and \((6, 23)\):
[tex]\[ \text{Slope}_3 = \frac{23 - 15}{6 - 4} = \frac{8}{2} = 4 \][/tex]
Since the calculated slopes between each pair of consecutive points are equal, this indicates that function \(B\) is indeed a linear function with a constant rate of change. Therefore, the rate of change of function \(B\) can be considered as the average of these slopes, which, in this case, is simply 4 because all slopes are the same.
Hence, the rate of change of function \(B\) is:
[tex]\[ \text{Rate of Change of Function } B = 4 \][/tex]
### Comparison
We now compare the rates of change of the two functions:
- The rate of change of function \(A\) is 2.
- The rate of change of function \(B\) is 4.
Thus, the correct statement is:
[tex]\[ \text{The rate of change of function } A \text{ is 2. The rate of change of function } B \text{ is 4.} \][/tex]
### Function \(A\)
Function \(A\) is represented by the equation \(y = 2x + 1\).
In this equation, the rate of change (slope) is given by the coefficient of \(x\). Therefore, the rate of change of function \(A\) is:
[tex]\[ \text{Rate of Change of Function } A = 2 \][/tex]
### Function \(B\)
Function \(B\) goes through the points \((1, 3)\), \((3, 11)\), \((4, 15)\), and \((6, 23)\).
To determine the rate of change for function \(B\), we calculate the slope between each pair of consecutive points:
1. Between \((1, 3)\) and \((3, 11)\):
[tex]\[ \text{Slope}_1 = \frac{11 - 3}{3 - 1} = \frac{8}{2} = 4 \][/tex]
2. Between \((3, 11)\) and \((4, 15)\):
[tex]\[ \text{Slope}_2 = \frac{15 - 11}{4 - 3} = \frac{4}{1} = 4 \][/tex]
3. Between \((4, 15)\) and \((6, 23)\):
[tex]\[ \text{Slope}_3 = \frac{23 - 15}{6 - 4} = \frac{8}{2} = 4 \][/tex]
Since the calculated slopes between each pair of consecutive points are equal, this indicates that function \(B\) is indeed a linear function with a constant rate of change. Therefore, the rate of change of function \(B\) can be considered as the average of these slopes, which, in this case, is simply 4 because all slopes are the same.
Hence, the rate of change of function \(B\) is:
[tex]\[ \text{Rate of Change of Function } B = 4 \][/tex]
### Comparison
We now compare the rates of change of the two functions:
- The rate of change of function \(A\) is 2.
- The rate of change of function \(B\) is 4.
Thus, the correct statement is:
[tex]\[ \text{The rate of change of function } A \text{ is 2. The rate of change of function } B \text{ is 4.} \][/tex]
