Answered

Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Our platform provides a seamless experience for finding reliable answers from a knowledgeable network of professionals. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.

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 :

GinnyAnswer
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 hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.