Answer :
To determine why the output values for linear function [tex]\( A \)[/tex] will always be different from the corresponding output values for linear function [tex]\( B \)[/tex], we need to analyze the given data and calculate the linear functions represented by [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
### Step-by-Step Solution:
1. Identify the input-output pairs for each function:
- For [tex]\( A \)[/tex]:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 3 \\ \hline 3 & 7 \\ \hline 5 & 11 \\ \hline 7 & 15 \\ \hline 9 & 19 \\ \hline \end{array} \][/tex]
- For [tex]\( B \)[/tex]:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 4 \\ \hline 3 & 8 \\ \hline 5 & 12 \\ \hline 7 & 16 \\ \hline 9 & 20 \\ \hline \end{array} \][/tex]
2. Calculate the slopes (rate of change) of the functions:
The slope formula for a linear function given two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
- For [tex]\( A \)[/tex], using points [tex]\((1, 3)\)[/tex] and [tex]\((3, 7)\)[/tex]:
[tex]\[ \text{slope}_A = \frac{7 - 3}{3 - 1} = \frac{4}{2} = 2 \][/tex]
- For [tex]\( B \)[/tex], using points [tex]\((1, 4)\)[/tex] and [tex]\((3, 8)\)[/tex]:
[tex]\[ \text{slope}_B = \frac{8 - 4}{3 - 1} = \frac{4}{2} = 2 \][/tex]
Hence, both functions [tex]\( A \)[/tex] and [tex]\( B \)[/tex] have the same slope of [tex]\( 2 \)[/tex].
3. Calculate the y-intercepts of the functions:
The y-intercept ([tex]\( b \)[/tex]) of a linear function [tex]\( y = mx + b \)[/tex] can be found by substituting one of the points and the slope into the equation.
- For [tex]\( A \)[/tex], using point [tex]\((1, 3)\)[/tex]:
[tex]\[ 3 = 2 \cdot 1 + b \implies 3 = 2 + b \implies b_A = 1 \][/tex]
- For [tex]\( B \)[/tex], using point [tex]\((1, 4)\)[/tex]:
[tex]\[ 4 = 2 \cdot 1 + b \implies 4 = 2 + b \implies b_B = 2 \][/tex]
Hence, the y-intercept of function [tex]\( A \)[/tex] is [tex]\( 1 \)[/tex] and for function [tex]\( B \)[/tex] it is [tex]\( 2 \)[/tex].
4. Compare the slopes and y-intercepts:
- The slopes of both functions [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are equal ([tex]\( 2 \)[/tex]).
- The y-intercepts of the functions are different: [tex]\( 1 \)[/tex] for [tex]\( A \)[/tex] and [tex]\( 2 \)[/tex] for [tex]\( B \)[/tex].
### Conclusion:
The output values for function [tex]\( A \)[/tex] will always be different from the corresponding output values for function [tex]\( B \)[/tex] because although they have the same rate of change (slope), their initial values (y-intercepts) are different.
Thus, the correct statement is:
- The initial values of the two functions are different, and the rates of change of the two functions are the same.
### Answer:
The initial values of the two functions are different, and the rates of change of the two functions are the same.
### Step-by-Step Solution:
1. Identify the input-output pairs for each function:
- For [tex]\( A \)[/tex]:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 3 \\ \hline 3 & 7 \\ \hline 5 & 11 \\ \hline 7 & 15 \\ \hline 9 & 19 \\ \hline \end{array} \][/tex]
- For [tex]\( B \)[/tex]:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 4 \\ \hline 3 & 8 \\ \hline 5 & 12 \\ \hline 7 & 16 \\ \hline 9 & 20 \\ \hline \end{array} \][/tex]
2. Calculate the slopes (rate of change) of the functions:
The slope formula for a linear function given two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
- For [tex]\( A \)[/tex], using points [tex]\((1, 3)\)[/tex] and [tex]\((3, 7)\)[/tex]:
[tex]\[ \text{slope}_A = \frac{7 - 3}{3 - 1} = \frac{4}{2} = 2 \][/tex]
- For [tex]\( B \)[/tex], using points [tex]\((1, 4)\)[/tex] and [tex]\((3, 8)\)[/tex]:
[tex]\[ \text{slope}_B = \frac{8 - 4}{3 - 1} = \frac{4}{2} = 2 \][/tex]
Hence, both functions [tex]\( A \)[/tex] and [tex]\( B \)[/tex] have the same slope of [tex]\( 2 \)[/tex].
3. Calculate the y-intercepts of the functions:
The y-intercept ([tex]\( b \)[/tex]) of a linear function [tex]\( y = mx + b \)[/tex] can be found by substituting one of the points and the slope into the equation.
- For [tex]\( A \)[/tex], using point [tex]\((1, 3)\)[/tex]:
[tex]\[ 3 = 2 \cdot 1 + b \implies 3 = 2 + b \implies b_A = 1 \][/tex]
- For [tex]\( B \)[/tex], using point [tex]\((1, 4)\)[/tex]:
[tex]\[ 4 = 2 \cdot 1 + b \implies 4 = 2 + b \implies b_B = 2 \][/tex]
Hence, the y-intercept of function [tex]\( A \)[/tex] is [tex]\( 1 \)[/tex] and for function [tex]\( B \)[/tex] it is [tex]\( 2 \)[/tex].
4. Compare the slopes and y-intercepts:
- The slopes of both functions [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are equal ([tex]\( 2 \)[/tex]).
- The y-intercepts of the functions are different: [tex]\( 1 \)[/tex] for [tex]\( A \)[/tex] and [tex]\( 2 \)[/tex] for [tex]\( B \)[/tex].
### Conclusion:
The output values for function [tex]\( A \)[/tex] will always be different from the corresponding output values for function [tex]\( B \)[/tex] because although they have the same rate of change (slope), their initial values (y-intercepts) are different.
Thus, the correct statement is:
- The initial values of the two functions are different, and the rates of change of the two functions are the same.
### Answer:
The initial values of the two functions are different, and the rates of change of the two functions are the same.
