Question
How does the graph show the change from line W to line X. the graph shows line W going from points (10, 10) and (-10, -10) and Line X going from points (9, -3) and (-9, 3)
a. the slope is changed by a factor of 3.
b. there is a slope change from positive to negative and by a factor of 3.
c. there is a slope change from positive to negative and no factor change.
d.there is a slope change from negative to positive and by a factor of 3.
a. the slope is changed by a factor of 3.
b. there is a slope change from positive to negative and by a factor of 3.
c. there is a slope change from positive to negative and no factor change.
d.there is a slope change from negative to positive and by a factor of 3.
Answers
GPT-4o mini
To analyze the change from Line W to Line X based on the given points, we first need to find the slopes of both lines.
**Line W:**
- Points: (10, 10) and (-10, -10)
- Slope (m) of Line W can be calculated using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
- Using points (10, 10) and (-10, -10):
\[
m_W = \frac{-10 - 10}{-10 - 10} = \frac{-20}{-20} = 1
\]
**Line X:**
- Points: (9, -3) and (-9, 3)
- Using the same slope formula:
\[
m_X = \frac{3 - (-3)}{-9 - 9} = \frac{3 + 3}{-18} = \frac{6}{-18} = -\frac{1}{3}
\]
Now, we can compare the slopes of the two lines:
- Slope of Line W (m_W) = 1 (positive slope)
- Slope of Line X (m_X) = -\(\frac{1}{3}\) (negative slope)
Now, let's analyze the options:
a. **The slope is changed by a factor of 3.**
- The absolute value of the slopes:
- From 1 (Line W) to \(-\frac{1}{3}\) (Line X) does show a change, but it is not simply the factor of 3 because \( \frac{1}{3} \) is not equal to 3.
b. **There is a slope change from positive to negative and by a factor of 3.**
- This is also incorrect. While there is a change from positive to negative, the factor is not 3.
c. **There is a slope change from positive to negative and no factor change.**
- This is correct regarding the slopes (positive to negative), but there is a factor change from 1 to \(-\frac{1}{3}\), as it isn't simply "no factor change."
d. **There is a slope change from negative to positive and by a factor of 3.**
- This is incorrect since the change is from positive (1) to negative (\(-\frac{1}{3}\)).
The correct conclusion regarding the change is:
- The slopes changed from a positive value of 1 to a negative value of -\(\frac{1}{3}\), indicating a change in direction from positive to negative, but the statement about "by a factor of 3" is not accurate.
Therefore, none of the options provided is entirely correct. However, option **B** somewhat conveys the change: there is a slope change from positive to negative, but the factor isn't accurately stated.
Hence, based on what's provided, **b** may closely represent the change in direction even if the factor is incorrectly expressed.
**Line W:**
- Points: (10, 10) and (-10, -10)
- Slope (m) of Line W can be calculated using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
- Using points (10, 10) and (-10, -10):
\[
m_W = \frac{-10 - 10}{-10 - 10} = \frac{-20}{-20} = 1
\]
**Line X:**
- Points: (9, -3) and (-9, 3)
- Using the same slope formula:
\[
m_X = \frac{3 - (-3)}{-9 - 9} = \frac{3 + 3}{-18} = \frac{6}{-18} = -\frac{1}{3}
\]
Now, we can compare the slopes of the two lines:
- Slope of Line W (m_W) = 1 (positive slope)
- Slope of Line X (m_X) = -\(\frac{1}{3}\) (negative slope)
Now, let's analyze the options:
a. **The slope is changed by a factor of 3.**
- The absolute value of the slopes:
- From 1 (Line W) to \(-\frac{1}{3}\) (Line X) does show a change, but it is not simply the factor of 3 because \( \frac{1}{3} \) is not equal to 3.
b. **There is a slope change from positive to negative and by a factor of 3.**
- This is also incorrect. While there is a change from positive to negative, the factor is not 3.
c. **There is a slope change from positive to negative and no factor change.**
- This is correct regarding the slopes (positive to negative), but there is a factor change from 1 to \(-\frac{1}{3}\), as it isn't simply "no factor change."
d. **There is a slope change from negative to positive and by a factor of 3.**
- This is incorrect since the change is from positive (1) to negative (\(-\frac{1}{3}\)).
The correct conclusion regarding the change is:
- The slopes changed from a positive value of 1 to a negative value of -\(\frac{1}{3}\), indicating a change in direction from positive to negative, but the statement about "by a factor of 3" is not accurate.
Therefore, none of the options provided is entirely correct. However, option **B** somewhat conveys the change: there is a slope change from positive to negative, but the factor isn't accurately stated.
Hence, based on what's provided, **b** may closely represent the change in direction even if the factor is incorrectly expressed.