To determine which points lie on the graph of the equation [tex]\( y = 1.5 + \lceil x \rceil \)[/tex], we need to understand the ceiling function, [tex]\( \lceil x \rceil \)[/tex]. The ceiling function, [tex]\( \lceil x \rceil \)[/tex], rounds a number [tex]\( x \)[/tex] up to the nearest integer. Using this, let's check each point:
1. Point [tex]\((-4.5, -2.5)\)[/tex]:
- Calculate [tex]\( \lceil -4.5 \rceil \)[/tex]:
[tex]\[ \lceil -4.5 \rceil = -4 \][/tex]
- Substitute [tex]\( x = -4.5 \)[/tex] into the equation:
[tex]\[ y = 1.5 + \lceil -4.5 \rceil = 1.5 + (-4) = -2.5 \][/tex]
- The [tex]\( y \)[/tex]-coordinate matches, so [tex]\((-4.5, -2.5)\)[/tex] is on the graph.
2. Point [tex]\((-0.8, 0.5)\)[/tex]:
- Calculate [tex]\( \lceil -0.8 \rceil \)[/tex]:
[tex]\[ \lceil -0.8 \rceil = 0 \][/tex]
- Substitute [tex]\( x = -0.8 \)[/tex] into the equation:
[tex]\[ y = 1.5 + \lceil -0.8 \rceil = 1.5 + 0 = 1.5 \][/tex]
- The [tex]\( y \)[/tex]-coordinate does not match, so [tex]\((-0.8, 0.5)\)[/tex] is not on the graph.
3. Point [tex]\((7.9, 9.5)\)[/tex]:
- Calculate [tex]\( \lceil 7.9 \rceil \)[/tex]:
[tex]\[ \lceil 7.9 \rceil = 8 \][/tex]
- Substitute [tex]\( x = 7.9 \)[/tex] into the equation:
[tex]\[ y = 1.5 + \lceil 7.9 \rceil = 1.5 + 8 = 9.5 \][/tex]
- The [tex]\( y \)[/tex]-coordinate matches, so [tex]\((7.9, 9.5)\)[/tex] is on the graph.
4. Point [tex]\((4.5, 6)\)[/tex]:
- Calculate [tex]\( \lceil 4.5 \rceil \)[/tex]:
[tex]\[ \lceil 4.5 \rceil = 5 \][/tex]
- Substitute [tex]\( x = 4.5 \)[/tex] into the equation:
[tex]\[ y = 1.5 + \lceil 4.5 \rceil = 1.5 + 5 = 6.5 \][/tex]
- The [tex]\( y \)[/tex]-coordinate does not match, so [tex]\((4.5, 6)\)[/tex] is not on the graph.
5. Point [tex]\((1.3, 3.5)\)[/tex]:
- Calculate [tex]\( \lceil 1.3 \rceil \)[/tex]:
[tex]\[ \lceil 1.3 \rceil = 2 \][/tex]
- Substitute [tex]\( x = 1.3 \)[/tex] into the equation:
[tex]\[ y = 1.5 + \lceil 1.3 \rceil = 1.5 + 2 = 3.5 \][/tex]
- The [tex]\( y \)[/tex]-coordinate matches, so [tex]\((1.3, 3.5)\)[/tex] is on the graph.
Therefore, the points on the graph are:
- [tex]\((-4.5, -2.5)\)[/tex]
- [tex]\((7.9, 9.5)\)[/tex]
- [tex]\((1.3, 3.5)\)[/tex]
