Discover new knowledge and insights with IDNLearn.com's extensive Q&A database. Find in-depth and accurate answers to all your questions from our knowledgeable and dedicated community members.
Sagot :
To solve this problem, we need to determine the finite differences and assess the type of function that the data models. Let's proceed step-by-step.
### Step 1: Calculate the First Differences
The first differences are computed by subtracting each [tex]\( y \)[/tex]-value from the subsequent [tex]\( y \)[/tex]-value in the table.
1. From [tex]\( y_1 = 21 \)[/tex] to [tex]\( y_2 = 11 \)[/tex]:
[tex]\[ 11 - 21 = -10 \][/tex]
2. From [tex]\( y_2 = 11 \)[/tex] to [tex]\( y_3 = 5 \)[/tex]:
[tex]\[ 5 - 11 = -6 \][/tex]
3. From [tex]\( y_3 = 5 \)[/tex] to [tex]\( y_4 = 3 \)[/tex]:
[tex]\[ 3 - 5 = -2 \][/tex]
4. From [tex]\( y_4 = 3 \)[/tex] to [tex]\( y_5 = 5 \)[/tex]:
[tex]\[ 5 - 3 = 2 \][/tex]
So, the first differences are:
[tex]\[ [-10, -6, -2, 2] \][/tex]
### Step 2: Calculate the Second Differences
The second differences are calculated by subtracting each first difference from the subsequent first difference.
1. From [tex]\(-10\)[/tex] to [tex]\(-6\)[/tex]:
[tex]\[ -6 - (-10) = -6 + 10 = 4 \][/tex]
2. From [tex]\(-6\)[/tex] to [tex]\(-2\)[/tex]:
[tex]\[ -2 - (-6) = -2 + 6 = 4 \][/tex]
3. From [tex]\(-2\)[/tex] to [tex]\(2\)[/tex]:
[tex]\[ 2 - (-2) = 2 + 2 = 4 \][/tex]
So, the second differences are:
[tex]\[ [4, 4, 4] \][/tex]
### Step 3: Organize the Results in the Table
Now we can fill in the table with our computed differences:
\begin{tabular}{|c|c|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] & \begin{tabular}{l}[tex]$1^{\text {st }}$[/tex] \\ difference\end{tabular} & \begin{tabular}{l}[tex]$2^{\text {nd }}$[/tex] \\ difference\end{tabular} \\
\hline-1 & 21 & -10 & \\
\hline 0 & 11 & -6 & 4 \\
\hline 1 & 5 & -2 & 4 \\
\hline 2 & 3 & 2 & 4 \\
\hline 3 & 5 & & \\
\hline
\end{tabular}
### Step 4: Analyze the Function Type
The second differences are constant and equal to 4. When the second differences are constant, this indicates that the function modeled by the data is a quadratic function.
[tex]\[ \boxed{\text{The table models a quadratic function because the second differences are constant.}} \][/tex]
### Step 1: Calculate the First Differences
The first differences are computed by subtracting each [tex]\( y \)[/tex]-value from the subsequent [tex]\( y \)[/tex]-value in the table.
1. From [tex]\( y_1 = 21 \)[/tex] to [tex]\( y_2 = 11 \)[/tex]:
[tex]\[ 11 - 21 = -10 \][/tex]
2. From [tex]\( y_2 = 11 \)[/tex] to [tex]\( y_3 = 5 \)[/tex]:
[tex]\[ 5 - 11 = -6 \][/tex]
3. From [tex]\( y_3 = 5 \)[/tex] to [tex]\( y_4 = 3 \)[/tex]:
[tex]\[ 3 - 5 = -2 \][/tex]
4. From [tex]\( y_4 = 3 \)[/tex] to [tex]\( y_5 = 5 \)[/tex]:
[tex]\[ 5 - 3 = 2 \][/tex]
So, the first differences are:
[tex]\[ [-10, -6, -2, 2] \][/tex]
### Step 2: Calculate the Second Differences
The second differences are calculated by subtracting each first difference from the subsequent first difference.
1. From [tex]\(-10\)[/tex] to [tex]\(-6\)[/tex]:
[tex]\[ -6 - (-10) = -6 + 10 = 4 \][/tex]
2. From [tex]\(-6\)[/tex] to [tex]\(-2\)[/tex]:
[tex]\[ -2 - (-6) = -2 + 6 = 4 \][/tex]
3. From [tex]\(-2\)[/tex] to [tex]\(2\)[/tex]:
[tex]\[ 2 - (-2) = 2 + 2 = 4 \][/tex]
So, the second differences are:
[tex]\[ [4, 4, 4] \][/tex]
### Step 3: Organize the Results in the Table
Now we can fill in the table with our computed differences:
\begin{tabular}{|c|c|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] & \begin{tabular}{l}[tex]$1^{\text {st }}$[/tex] \\ difference\end{tabular} & \begin{tabular}{l}[tex]$2^{\text {nd }}$[/tex] \\ difference\end{tabular} \\
\hline-1 & 21 & -10 & \\
\hline 0 & 11 & -6 & 4 \\
\hline 1 & 5 & -2 & 4 \\
\hline 2 & 3 & 2 & 4 \\
\hline 3 & 5 & & \\
\hline
\end{tabular}
### Step 4: Analyze the Function Type
The second differences are constant and equal to 4. When the second differences are constant, this indicates that the function modeled by the data is a quadratic function.
[tex]\[ \boxed{\text{The table models a quadratic function because the second differences are constant.}} \][/tex]
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. For precise answers, trust IDNLearn.com. Thank you for visiting, and we look forward to helping you again soon.