Answer :
Let's carefully analyze the problem and the given choices to determine which statement best describes DJ's mistake.
First, we should identify the correct components of the equation [tex]\( y = -\frac{1}{2}x + 1 \)[/tex]:
- The y-intercept is 1. This means the line crosses the y-axis at the point (0, 1).
- The slope of the line is [tex]\(-\frac{1}{2}\)[/tex]. This indicates that for every unit the line moves to the right along the x-axis, it moves down by [tex]\(\frac{1}{2}\)[/tex] units along the y-axis, representing a negative rate of change.
Now let's examine the given statements one by one to identify DJ's mistake:
1. The y-intercept is plotted as a positive 1 when it should be plotted at negative 1.
- This statement is incorrect. The correct y-intercept from the equation [tex]\( y = -\frac{1}{2}x + 1 \)[/tex] is indeed positive 1, not negative 1.
2. The rate of change of the line is graphed as the reciprocal.
- This statement suggests that instead of using the slope [tex]\(-\frac{1}{2}\)[/tex], DJ used its reciprocal [tex]\(-2\)[/tex]. This is incorrect, as the equation gives the slope as [tex]\(-\frac{1}{2}\)[/tex].
3. The rate of change of the line is positive when it should be graphed as a negative rate of change.
- This statement suggests that DJ graphed the line with a positive slope instead of a negative one. Since the equation [tex]\( y = -\frac{1}{2}x + 1 \)[/tex] has a negative slope of [tex]\(-\frac{1}{2}\)[/tex], a mistake involving plotting a positive slope would imply DJ mistakenly made the line ascend as one moves to the right, contrary to the intended descent. This possibility accurately reflects a common mistake.
4. The y-intercept is plotted at 1 when it should be plotted at [tex]\(-\frac{1}{2}\)[/tex].
- This statement incorrectly suggests that the y-intercept should be [tex]\(-\frac{1}{2}\)[/tex] when it is actually 1.
Based on this analysis, the most appropriate statement indicating DJ's mistake would be:
The rate of change of the line is positive when it should be graphed as a negative rate of change.
Thus, the correct answer is:
3
First, we should identify the correct components of the equation [tex]\( y = -\frac{1}{2}x + 1 \)[/tex]:
- The y-intercept is 1. This means the line crosses the y-axis at the point (0, 1).
- The slope of the line is [tex]\(-\frac{1}{2}\)[/tex]. This indicates that for every unit the line moves to the right along the x-axis, it moves down by [tex]\(\frac{1}{2}\)[/tex] units along the y-axis, representing a negative rate of change.
Now let's examine the given statements one by one to identify DJ's mistake:
1. The y-intercept is plotted as a positive 1 when it should be plotted at negative 1.
- This statement is incorrect. The correct y-intercept from the equation [tex]\( y = -\frac{1}{2}x + 1 \)[/tex] is indeed positive 1, not negative 1.
2. The rate of change of the line is graphed as the reciprocal.
- This statement suggests that instead of using the slope [tex]\(-\frac{1}{2}\)[/tex], DJ used its reciprocal [tex]\(-2\)[/tex]. This is incorrect, as the equation gives the slope as [tex]\(-\frac{1}{2}\)[/tex].
3. The rate of change of the line is positive when it should be graphed as a negative rate of change.
- This statement suggests that DJ graphed the line with a positive slope instead of a negative one. Since the equation [tex]\( y = -\frac{1}{2}x + 1 \)[/tex] has a negative slope of [tex]\(-\frac{1}{2}\)[/tex], a mistake involving plotting a positive slope would imply DJ mistakenly made the line ascend as one moves to the right, contrary to the intended descent. This possibility accurately reflects a common mistake.
4. The y-intercept is plotted at 1 when it should be plotted at [tex]\(-\frac{1}{2}\)[/tex].
- This statement incorrectly suggests that the y-intercept should be [tex]\(-\frac{1}{2}\)[/tex] when it is actually 1.
Based on this analysis, the most appropriate statement indicating DJ's mistake would be:
The rate of change of the line is positive when it should be graphed as a negative rate of change.
Thus, the correct answer is:
3
