To find the equation of the quadratic function [tex]\( p(x) \)[/tex] based on the given table of values, the function should be in the vertex form [tex]\( p(x) = a(x - h)^2 + k \)[/tex], where [tex]\( (h, k) \)[/tex] is the vertex of the parabola, and [tex]\( a \)[/tex] is a coefficient that determines the width and direction of the parabola.
Given the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & p(x) \\
\hline
-1 & 31 \\
0 & 17 \\
1 & 7 \\
2 & 1 \\
3 & -1 \\
4 & 1 \\
5 & 7 \\
\hline
\end{array}
\][/tex]
Let's use the given answer directly to check and identify the correct function form.
### Checking [tex]\( p(x) = 2(x+3)^2 - 1 \)[/tex]:
#### Verifying [tex]\( p(x) \)[/tex] at various values of [tex]\( x \)[/tex]:
1. [tex]\( x = 3 \)[/tex]:
[tex]\[
p(3) = 2(3 + 3)^2 - 1 = 2 (6)^2 - 1 = 2 \cdot 36 - 1 = 72 - 1 = 71 \, , \text{(This doesn't match)}.
\][/tex]
This value is wrong. Hence, this form is incorrect.
### Verifying [tex]\( p(x) = 2(x-3)^2 - 1 \)[/tex]:
#### Verifying [tex]\( p(x) \)[/tex] at various values of [tex]\( x \)[/tex]:
1. [tex]\( x = -1 \)[/tex]:
[tex]\[
p(-1) = 2(-1 - 3)^2 - 1 = 2(-4)^2 - 1 = 2 \cdot 16 - 1 = 32 - 1 = 31 \, (\text{Correct!})
\][/tex]
2. [tex]\( x = 0 \)[/tex]:
[tex]\[
p(0) = 2(0 - 3)^2 - 1 = 2(-3)^2 - 1 = 2 \cdot 9 - 1 = 18 - 1 = 17 \, (\text{Correct!})
\][/tex]
3. [tex]\( x = 1 \)[/tex]:
[tex]\[
p(1) = 2(1 - 3)^2 - 1 = 2(-2)^2 - 1 = 2 \cdot 4 - 1 = 8 - 1 = 7 \, (\text{Correct!})
\][/tex]
4. [tex]\( x = 2 \)[/tex]:
[tex]\[
p(2) = 2(2 - 3)^2 - 1 = 2(-1)^2 - 1 = 2 \cdot 1 - 1 = 2 - 1 = 1 \, (\text{Correct!})
\][/tex]
5. [tex]\( x = 3 \)[/tex]:
[tex]\[
p(3) = 2(3 - 3)^2 - 1 = 2(0)^2 - 1 = 0 - 1 = -1 \, (\text{Correct!})
\][/tex]
6. [tex]\( x = 4 \)[/tex]:
[tex]\[
p(4) = 2(4 - 3)^2 - 1 = 2(1)^2 - 1 = 2 \cdot 1 - 1 = 2 - 1 = 1 \, (\text{Correct!})
\][/tex]
7. [tex]\( x = 5 \)[/tex]:
[tex]\[
p(5) = 2(5 - 3)^2 - 1 = 2(2)^2 - 1 = 2 \cdot 4 - 1 = 8 - 1 = 7 \, (\text{Correct!})
\][/tex]
Since all the values match, we confirm that the correct equation of [tex]\( p(x) \)[/tex] is [tex]\( 2(x-3)^2-1 \)[/tex].
### Conclusion:
The equation of [tex]\( p(x) \)[/tex] in vertex form is:
[tex]\[ p(x) = 2(x-3)^2 - 1 \][/tex]
