To determine the correct equation for [tex]\( p(x) \)[/tex] that fits the given data points [tex]\((-1, 10), (0, 1), (1, -2), (2, 1), (3, 10), (4, 25), (5, 46)\)[/tex], we will verify each of the provided equations by substituting the given [tex]\( x \)[/tex]-values and checking if the equation produces the corresponding [tex]\( p(x) \)[/tex]-values.
Given equations:
1. [tex]\( p(x) = 2(x-1)^2 - 2 \)[/tex]
2. [tex]\( p(x) = 2(x+1)^2 - 2 \)[/tex]
3. [tex]\( p(x) = 3(x-1)^2 - 2 \)[/tex]
4. [tex]\( p(x) = 3(x+1)^2 - 2 \)[/tex]
We need to find out which of these equations produces the correct [tex]\( p(x) \)[/tex] values for all given [tex]\( x \)[/tex]-values.
Firstly, let's try the equation [tex]\( p(x) = 3(x-1)^2 - 2 \)[/tex]:
For [tex]\( x = -1 \)[/tex]:
[tex]\[ p(-1) = 3(-1-1)^2 - 2 = 3(-2)^2 - 2 = 3 \cdot 4 - 2 = 12 - 2 = 10 \][/tex]
[tex]\( p(-1) = 10 \)[/tex] which matches the table.
For [tex]\( x = 0 \)[/tex]:
[tex]\[ p(0) = 3(0-1)^2 - 2 = 3(-1)^2 - 2 = 3 \cdot 1 - 2 = 3 - 2 = 1 \][/tex]
[tex]\( p(0) = 1 \)[/tex] which matches the table.
For [tex]\( x = 1 \)[/tex]:
[tex]\[ p(1) = 3(1-1)^2 - 2 = 3(0)^2 - 2 = 3 \cdot 0 - 2 = 0 - 2 = -2 \][/tex]
[tex]\( p(1) = -2 \)[/tex] which matches the table.
For [tex]\( x = 2 \)[/tex]:
[tex]\[ p(2) = 3(2-1)^2 - 2 = 3(1)^2 - 2 = 3 \cdot 1 - 2 = 3 - 2 = 1 \][/tex]
[tex]\( p(2) = 1 \)[/tex] which matches the table.
For [tex]\( x = 3 \)[/tex]:
[tex]\[ p(3) = 3(3-1)^2 - 2 = 3(2)^2 - 2 = 3 \cdot 4 - 2 = 12 - 2 = 10 \][/tex]
[tex]\( p(3) = 10 \)[/tex] which matches the table.
For [tex]\( x = 4 \)[/tex]:
[tex]\[ p(4) = 3(4-1)^2 - 2 = 3(3)^2 - 2 = 3 \cdot 9 - 2 = 27 - 2 = 25 \][/tex]
[tex]\( p(4) = 25 \)[/tex] which matches the table.
For [tex]\( x = 5 \)[/tex]:
[tex]\[ p(5) = 3(5-1)^2 - 2 = 3(4)^2 - 2 = 3 \cdot 16 - 2 = 48 - 2 = 46 \][/tex]
[tex]\( p(5) = 46 \)[/tex] which matches the table.
Since the function [tex]\( p(x) = 3(x-1)^2 - 2 \)[/tex] produces the correct [tex]\( p(x) \)[/tex] values for all given [tex]\( x \)[/tex]-values, the equation of [tex]\( p(x) \)[/tex] in vertex form is:
[tex]\[ p(x) = 3(x-1)^2 - 2 \][/tex]
