Let's evaluate the polynomial equation [tex]\(y=\frac{1}{2}x^3+2x^2+4x-6\)[/tex] for the given values of [tex]\(x\)[/tex].
### When [tex]\(x = 2\)[/tex]:
1. Start by substituting [tex]\(x = 2\)[/tex] into the polynomial:
[tex]\[
y = \frac{1}{2}(2)^3 + 2(2)^2 + 4(2) - 6
\][/tex]
2. Evaluate each term:
- [tex]\(\frac{1}{2}(2)^3 = \frac{1}{2} \times 8 = 4\)[/tex]
- [tex]\(2(2)^2 = 2 \times 4 = 8\)[/tex]
- [tex]\(4(2) = 8\)[/tex]
- The constant term is [tex]\(-6\)[/tex]
3. Add the evaluated terms together:
[tex]\[
y = 4 + 8 + 8 - 6 = 14
\][/tex]
Therefore, when [tex]\(x = 2\)[/tex], [tex]\(y = 14\)[/tex].
### When [tex]\(x = 3\)[/tex]:
1. Substitute [tex]\(x = 3\)[/tex] into the polynomial:
[tex]\[
y = \frac{1}{2}(3)^3 + 2(3)^2 + 4(3) - 6
\][/tex]
2. Evaluate each term:
- [tex]\(\frac{1}{2}(3)^3 = \frac{1}{2} \times 27 = 13.5\)[/tex]
- [tex]\(2(3)^2 = 2 \times 9 = 18\)[/tex]
- [tex]\(4(3) = 12\)[/tex]
- The constant term is [tex]\(-6\)[/tex]
3. Add the evaluated terms together:
[tex]\[
y = 13.5 + 18 + 12 - 6 = 37.5
\][/tex]
Therefore, when [tex]\(x = 3\)[/tex], [tex]\(y = 37.5\)[/tex].
### Final Answers:
- If [tex]\(x = 2\)[/tex], then [tex]\(y = 14\)[/tex].
- If [tex]\(x = 3\)[/tex], then [tex]\(y = 37.5\)[/tex].
