Let's analyze the given table of values for the function [tex]\( f \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 & 5 \\
\hline
f(x) & 13 & 19 & 37 & 91 & 253 \\
\hline
\end{array}
\][/tex]
### 1. Determining the Parent Function:
First, let's determine the type of function. By examining the values, we see that [tex]\( f \)[/tex] is neither linear nor quadratic; it grows faster than a quadratic function. Thus, it must be a higher-degree polynomial.
Therefore, the parent function of the function represented in the table is:
polynomial function.
### 2. Translating the Function Down 4 Units:
If function [tex]\( f \)[/tex] was translated down 4 units, each [tex]\( f(x) \)[/tex]-value decreases by 4.
So, let's compute the transformed [tex]\( f(x) \)[/tex]-values by subtracting 4 from each original [tex]\( f(x) \)[/tex]-value:
- [tex]\( 13 - 4 = 9 \)[/tex]
- [tex]\( 19 - 4 = 15 \)[/tex]
- [tex]\( 37 - 4 = 33 \)[/tex]
- [tex]\( 91 - 4 = 87 \)[/tex]
- [tex]\( 253 - 4 = 249 \)[/tex]
Thus, if function [tex]\( f \)[/tex] was translated down 4 units, the [tex]\( f(x) \)[/tex]-values would be:
array([9, 15, 33, 87, 249])
### 3. A Point in the Transformed Function's Table:
Lastly, let's consider the point at [tex]\( x = 1 \)[/tex]. The original point for [tex]\( x = 1 \)[/tex] was [tex]\( (1, 13) \)[/tex]. After translating the function down by 4 units, the new point becomes [tex]\( (1, 13 - 4) = (1, 9) \)[/tex].
So, a point in the table for the transformed function would be:
(1, 9)
Putting it all together:
- The parent function of the function represented in the table is polynomial function.
- If function [tex]\( f \)[/tex] was translated down 4 units, the [tex]\( f(x) \)[/tex]-values would be 9, 15, 33, 87, 249
- A point in the table for the transformed function would be (1, 9)
