Answered

Find expert advice and community support for all your questions on IDNLearn.com. Discover thorough and trustworthy answers from our community of knowledgeable professionals, tailored to meet your specific needs.

Given the table:

[tex]\[
\begin{tabular}{|l|l|}
\hline
$x$ & $p(x)$ \\
\hline
-1 & 10 \\
\hline
0 & 1 \\
\hline
1 & -2 \\
\hline
2 & 1 \\
\hline
3 & 10 \\
\hline
4 & 25 \\
\hline
5 & 46 \\
\hline
\end{tabular}
\][/tex]

What is the equation of [tex]\( p(x) \)[/tex] in vertex form?

A. [tex]\( p(x) = 2(x-1)^2 - 2 \)[/tex]
B. [tex]\( p(x) = 2(x+1)^2 - 2 \)[/tex]
C. [tex]\( p(x) = 3(x-1)^2 - 2 \)[/tex]
D. [tex]\( p(x) = 3(x+1)^2 - 2 \)[/tex]

Sagot :

GinnyAnsweridn
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]
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Find clear and concise answers at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.