To determine whether the given points lie on the graph of the linear function [tex]\( y = 2x + 3 \)[/tex], we will substitute the [tex]\( x \)[/tex]-coordinate of each point into the equation and check if the resulting [tex]\( y \)[/tex]-value matches the [tex]\( y \)[/tex]-coordinate of the point.
Let's analyze each point one by one:
### Point [tex]\((-1, 1)\)[/tex]
1. Substitute [tex]\( x = -1 \)[/tex] into the equation [tex]\( y = 2x + 3 \)[/tex]:
[tex]\[
y = 2(-1) + 3 = -2 + 3 = 1
\][/tex]
2. The [tex]\( y \)[/tex]-value we calculated is [tex]\( 1 \)[/tex], which matches the [tex]\( y \)[/tex]-coordinate of the point [tex]\((-1, 1)\)[/tex].
Thus, the point [tex]\((-1, 1)\)[/tex] lies on the graph.
### Point [tex]\((-4, -6)\)[/tex]
1. Substitute [tex]\( x = -4 \)[/tex] into the equation [tex]\( y = 2x + 3 \)[/tex]:
[tex]\[
y = 2(-4) + 3 = -8 + 3 = -5
\][/tex]
2. The [tex]\( y \)[/tex]-value we calculated is [tex]\( -5 \)[/tex], which does not match the [tex]\( y \)[/tex]-coordinate of the point [tex]\((-4, -6)\)[/tex].
Thus, the point [tex]\((-4, -6)\)[/tex] does not lie on the graph.
### Point [tex]\((0, -2)\)[/tex]
1. Substitute [tex]\( x = 0 \)[/tex] into the equation [tex]\( y = 2x + 3 \)[/tex]:
[tex]\[
y = 2(0) + 3 = 0 + 3 = 3
\][/tex]
2. The [tex]\( y \)[/tex]-value we calculated is [tex]\( 3 \)[/tex], which does not match the [tex]\( y \)[/tex]-coordinate of the point [tex]\((0, -2)\)[/tex].
Thus, the point [tex]\((0, -2)\)[/tex] does not lie on the graph.
### Point [tex]\((-2, -1)\)[/tex]
1. Substitute [tex]\( x = -2 \)[/tex] into the equation [tex]\( y = 2x + 3 \)[/tex]:
[tex]\[
y = 2(-2) + 3 = -4 + 3 = -1
\][/tex]
2. The [tex]\( y \)[/tex]-value we calculated is [tex]\( -1 \)[/tex], which matches the [tex]\( y \)[/tex]-coordinate of the point [tex]\((-2, -1)\)[/tex].
Thus, the point [tex]\((-2, -1)\)[/tex] lies on the graph.
### Conclusion
After checking each point, we find that the points [tex]\((-1, 1)\)[/tex] and [tex]\((-2, -1)\)[/tex] lie on the graph of the function [tex]\( y = 2x + 3 \)[/tex].
