To determine which points lie on the graph of [tex]\( y = 1.5 + \lceil x \rceil \)[/tex], we need to evaluate each option. The function [tex]\(\lceil x \rceil\)[/tex] refers to the ceiling function, which rounds [tex]\(x\)[/tex] up to the nearest integer.
Let's evaluate each point:
1. Point [tex]\((-4.5, -2.5)\)[/tex]:
- Calculate [tex]\(\lceil -4.5 \rceil\)[/tex]. The ceiling of [tex]\(-4.5\)[/tex] is [tex]\(-4\)[/tex].
- Calculate [tex]\( 1.5 + \lceil -4.5 \rceil = 1.5 + (-4) = -2.5 \)[/tex].
- The point satisfies the equation because [tex]\( y = -2.5 \)[/tex].
2. Point [tex]\((-0.8, 0.5)\)[/tex]:
- Calculate [tex]\(\lceil -0.8 \rceil\)[/tex]. The ceiling of [tex]\(-0.8\)[/tex] is [tex]\(0 \)[/tex].
- Calculate [tex]\( 1.5 + \lceil -0.8 \rceil = 1.5 + 0 = 1.5 \)[/tex].
- The point does not satisfy the equation because [tex]\( y \neq 0.5 \)[/tex].
3. Point [tex]\((7.9, 9.5)\)[/tex]:
- Calculate [tex]\(\lceil 7.9 \rceil\)[/tex]. The ceiling of [tex]\( 7.9 \)[/tex] is [tex]\( 8 \)[/tex].
- Calculate [tex]\( 1.5 + \lceil 7.9 \rceil = 1.5 + 8 = 9.5 \)[/tex].
- The point satisfies the equation because [tex]\( y = 9.5 \)[/tex].
4. Point [tex]\((4.5, 6)\)[/tex]:
- Calculate [tex]\(\lceil 4.5 \rceil\)[/tex]. The ceiling of [tex]\( 4.5 \)[/tex] is [tex]\( 5 \)[/tex].
- Calculate [tex]\( 1.5 + \lceil 4.5 \rceil = 1.5 + 5 = 6.5 \)[/tex].
- The point does not satisfy the equation because [tex]\( y \neq 6 \)[/tex].
5. Point [tex]\((1.3, 3.5)\)[/tex]:
- Calculate [tex]\(\lceil 1.3 \rceil\)[/tex]. The ceiling of [tex]\( 1.3 \)[/tex] is [tex]\( 2 \)[/tex].
- Calculate [tex]\( 1.5 + \lceil 1.3 \rceil = 1.5 + 2 = 3.5 \)[/tex].
- The point satisfies the equation because [tex]\( y = 3.5 \)[/tex].
Therefore, the points that lie on the graph of [tex]\( y = 1.5 + \lceil x \rceil \)[/tex] are:
1. [tex]\((-4.5, -2.5)\)[/tex]
2. [tex]\((7.9, 9.5)\)[/tex]
3. [tex]\((1.3, 3.5)\)[/tex]
