To solve this problem, we need to determine the parent function based on the given table values, then determine the values after translating the function up by 5 units, and finally provide a point in the table for the transformed function.
1. Determining the Parent Function
The given table is:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 & 5 \\
\hline
f(x) & 9 & 11 & 15 & 23 & 39 \\
\hline
\end{array}
\][/tex]
From this table, we can observe that the values of [tex]\(f(x)\)[/tex] follow a certain pattern. Based on this pattern, the parent function can be identified as:
[tex]\[
x^2 + x + 7
\][/tex]
2. Translating the Function Up by 5 Units
When we translate the function [tex]\(f(x)\)[/tex] up by 5 units, we need to add 5 to each of the [tex]\(f(x)\)[/tex] values. Thus, we need to calculate the new values as follows:
[tex]\[
\begin{array}{ccc}
x & f(x) & f(x) + 5 \\
1 & 9 & 14 \\
2 & 11 & 16 \\
3 & 15 & 20 \\
4 & 23 & 28 \\
5 & 39 & 44 \\
\end{array}
\][/tex]
The [tex]\(y\)[/tex]-values after translating up 5 units are:
[tex]\[
[14, 16, 20, 28, 44]
\][/tex]
3. Providing a Point for the Transformed Function
For a specific point, consider [tex]\(x = 1\)[/tex]. The translated value for [tex]\(x = 1\)[/tex] is:
[tex]\[
f(1) + 5 = 9 + 5 = 14
\][/tex]
Therefore, the point in the table for the transformed function is:
[tex]\[
(1, 14)
\][/tex]
Combining all steps, we can complete the statements as follows:
- The parent function of the function represented in the table is [tex]\( \boldsymbol{x^2 + x + 7} \)[/tex].
- If function [tex]\(f\)[/tex] was translated up 5 units, the [tex]\( \boldsymbol{y} \)[/tex]-values would be [tex]\( \boldsymbol{[14, 16, 20, 28, 44]} \)[/tex].
- A point in the table for the transformed function would be [tex]\( \boldsymbol{(1, 14)} \)[/tex].
