Question
Compare Linear function 1 with linear function 2. Which one has the greater rate of change? Choose 1 for Linear Function 1; choose 2 for Linear Function 2.
Linear Function 1: y = x + 8
Linear function 2: a graph with (4 , 10) and (2 , 6)
Linear function ___ has the greater rate of change
Linear Function 1: y = x + 8
Linear function 2: a graph with (4 , 10) and (2 , 6)
Linear function ___ has the greater rate of change
Answers
Answered by
GPT-4o mini
To compare the rates of change of the two linear functions, we can determine the slope of each function.
**Linear Function 1:**
The function is given as \( y = x + 8 \). The slope (rate of change) of this function is 1, since it can be expressed in the form \( y = mx + b \) where \( m \) is the slope.
**Linear Function 2:**
For the second function, we can find the slope using the two points \((4, 10)\) and \((2, 6)\). The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting in the values from the points:
\[
\text{slope} = \frac{6 - 10}{2 - 4} = \frac{-4}{-2} = 2
\]
Now we can compare the slopes:
- Rate of change (slope) of Linear Function 1: 1
- Rate of change (slope) of Linear Function 2: 2
Since 2 > 1, Linear Function 2 has the greater rate of change.
Therefore:
Linear function **2** has the greater rate of change.
**Linear Function 1:**
The function is given as \( y = x + 8 \). The slope (rate of change) of this function is 1, since it can be expressed in the form \( y = mx + b \) where \( m \) is the slope.
**Linear Function 2:**
For the second function, we can find the slope using the two points \((4, 10)\) and \((2, 6)\). The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting in the values from the points:
\[
\text{slope} = \frac{6 - 10}{2 - 4} = \frac{-4}{-2} = 2
\]
Now we can compare the slopes:
- Rate of change (slope) of Linear Function 1: 1
- Rate of change (slope) of Linear Function 2: 2
Since 2 > 1, Linear Function 2 has the greater rate of change.
Therefore:
Linear function **2** has the greater rate of change.
Answer
you sure
Answered by
GPT-4o mini
Let’s review the calculations again to ensure accuracy.
**Linear Function 1:**
The function is given as \( y = x + 8 \). The slope (rate of change) is the coefficient of \( x \), which is \( 1 \).
**Linear Function 2:**
We have two points: \((4, 10)\) and \((2, 6)\). To find the slope, we apply the formula:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the points \((x_1, y_1) = (4, 10)\) and \((x_2, y_2) = (2, 6)\):
\[
\text{slope} = \frac{6 - 10}{2 - 4} = \frac{-4}{-2} = 2
\]
Thus, the rate of change for Linear Function 2 is indeed \( 2 \).
Now comparing the two slopes:
- Linear Function 1 (slope): \( 1 \)
- Linear Function 2 (slope): \( 2 \)
Since \( 2 > 1 \), it confirms that Linear Function 2 has the greater rate of change.
So, my original conclusion stands correct:
**Linear function 2 has the greater rate of change.**
**Linear Function 1:**
The function is given as \( y = x + 8 \). The slope (rate of change) is the coefficient of \( x \), which is \( 1 \).
**Linear Function 2:**
We have two points: \((4, 10)\) and \((2, 6)\). To find the slope, we apply the formula:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the points \((x_1, y_1) = (4, 10)\) and \((x_2, y_2) = (2, 6)\):
\[
\text{slope} = \frac{6 - 10}{2 - 4} = \frac{-4}{-2} = 2
\]
Thus, the rate of change for Linear Function 2 is indeed \( 2 \).
Now comparing the two slopes:
- Linear Function 1 (slope): \( 1 \)
- Linear Function 2 (slope): \( 2 \)
Since \( 2 > 1 \), it confirms that Linear Function 2 has the greater rate of change.
So, my original conclusion stands correct:
**Linear function 2 has the greater rate of change.**