Answer :
Let's complete the function table for the given domain, evaluating the function \( f(x) = (x-5)^2 + 1 \) at each provided value of \( x \).
We have:
[tex]\[ \begin{aligned} &f(2) = (2 - 5)^2 + 1 = (-3)^2 + 1 = 9 + 1 = 10, \\ &f(3) = (3 - 5)^2 + 1 = (-2)^2 + 1 = 4 + 1 = 5, \\ &f(4) = (4 - 5)^2 + 1 = (-1)^2 + 1 = 1 + 1 = 2, \\ &f(5) = (5 - 5)^2 + 1 = 0^2 + 1 = 0 + 1 = 1, \\ &f(6) = (6 - 5)^2 + 1 = 1^2 + 1 = 1 + 1 = 2. \end{aligned} \][/tex]
Let's fill in the table with these values:
[tex]\[ \begin{tabular}{|c|c|c|c|c|c|} \hline [tex]$x$[/tex] & 2 & 3 & 4 & 5 & 6 \\
\hline
[tex]$f(x)$[/tex] & 10 & 5 & 2 & 1 & 2 \\
\hline
\end{tabular}
\][/tex]
Next, to plot these points on a graph, we will mark each \( (x, f(x)) \) pair:
- (2, 10)
- (3, 5)
- (4, 2)
- (5, 1)
- (6, 2)
Here's how you can visualize the points on the graph:
1. (2, 10): Locate \( x = 2 \) on the horizontal axis and \( y = 10 \) on the vertical axis.
2. (3, 5): Locate \( x = 3 \) on the horizontal axis and \( y = 5 \) on the vertical axis.
3. (4, 2): Locate \( x = 4 \) on the horizontal axis and \( y = 2 \) on the vertical axis.
4. (5, 1): Locate \( x = 5 \) on the horizontal axis and \( y = 1 \) on the vertical axis.
5. (6, 2): Locate \( x = 6 \) on the horizontal axis and \( y = 2 \) on the vertical axis.
Connect these points to see the shape of the function [tex]\( f(x) \)[/tex].
We have:
[tex]\[ \begin{aligned} &f(2) = (2 - 5)^2 + 1 = (-3)^2 + 1 = 9 + 1 = 10, \\ &f(3) = (3 - 5)^2 + 1 = (-2)^2 + 1 = 4 + 1 = 5, \\ &f(4) = (4 - 5)^2 + 1 = (-1)^2 + 1 = 1 + 1 = 2, \\ &f(5) = (5 - 5)^2 + 1 = 0^2 + 1 = 0 + 1 = 1, \\ &f(6) = (6 - 5)^2 + 1 = 1^2 + 1 = 1 + 1 = 2. \end{aligned} \][/tex]
Let's fill in the table with these values:
[tex]\[ \begin{tabular}{|c|c|c|c|c|c|} \hline [tex]$x$[/tex] & 2 & 3 & 4 & 5 & 6 \\
\hline
[tex]$f(x)$[/tex] & 10 & 5 & 2 & 1 & 2 \\
\hline
\end{tabular}
\][/tex]
Next, to plot these points on a graph, we will mark each \( (x, f(x)) \) pair:
- (2, 10)
- (3, 5)
- (4, 2)
- (5, 1)
- (6, 2)
Here's how you can visualize the points on the graph:
1. (2, 10): Locate \( x = 2 \) on the horizontal axis and \( y = 10 \) on the vertical axis.
2. (3, 5): Locate \( x = 3 \) on the horizontal axis and \( y = 5 \) on the vertical axis.
3. (4, 2): Locate \( x = 4 \) on the horizontal axis and \( y = 2 \) on the vertical axis.
4. (5, 1): Locate \( x = 5 \) on the horizontal axis and \( y = 1 \) on the vertical axis.
5. (6, 2): Locate \( x = 6 \) on the horizontal axis and \( y = 2 \) on the vertical axis.
Connect these points to see the shape of the function [tex]\( f(x) \)[/tex].
