To determine in which quadrant or on which axis each of the points [tex]$(-2, 4)$[/tex], [tex]$(3, 1)$[/tex], [tex]$(-1, 0)$[/tex], and [tex]$(-3, -5)$[/tex] lie, we need to analyze the coordinates of each point individually based on the signs of their [tex]$x$[/tex] (horizontal) and [tex]$y$[/tex] (vertical) values.
1. Point [tex]$(-2, 4)$[/tex]:
- The [tex]$x$[/tex]-coordinate is [tex]$-2$[/tex], which is negative.
- The [tex]$y$[/tex]-coordinate is [tex]$4$[/tex], which is positive.
- A point with a negative [tex]$x$[/tex]-coordinate and a positive [tex]$y$[/tex]-coordinate lies in Quadrant II.
2. Point [tex]$(3, 1)$[/tex]:
- The [tex]$x$[/tex]-coordinate is [tex]$3$[/tex], which is positive.
- The [tex]$y$[/tex]-coordinate is [tex]$1$[/tex], which is positive.
- A point with both [tex]$x$[/tex] and [tex]$y$[/tex] coordinates positive lies in Quadrant I.
3. Point [tex]$(-1, 0)$[/tex]:
- The [tex]$x$[/tex]-coordinate is [tex]$-1$[/tex], which is negative.
- The [tex]$y$[/tex]-coordinate is [tex]$0$[/tex].
- A point with the [tex]$y$[/tex]-coordinate equal to [tex]$0$[/tex] and the [tex]$x$[/tex]-coordinate not equal to [tex]$0$[/tex] lies on the x-axis.
4. Point [tex]$(-3, -5)$[/tex]:
- The [tex]$x$[/tex]-coordinate is [tex]$-3$[/tex], which is negative.
- The [tex]$y$[/tex]-coordinate is [tex]$-5$[/tex], which is negative.
- A point with both [tex]$x$[/tex] and [tex]$y$[/tex] coordinates negative lies in Quadrant III.
Therefore, the points lie in the following quadrants or on the specified axis:
- [tex]$(-2, 4)$[/tex] is in Quadrant II.
- [tex]$(3, 1)$[/tex] is in Quadrant I.
- [tex]$(-1, 0)$[/tex] is on the x-axis.
- [tex]$(-3, -5)$[/tex] is in Quadrant III.
