Select the correct answer from each drop-down menu.
The function $f$ is given by the table of values as shown below.
\begin{tabular}{|c|c|c|c|c|c|}
\hline$x$ & 1 & 2 & 3 & 4 & 5 \\
\hline$f(x)$ & 13 & 19 & 37 & 91 & 253 \\
\hline
\end{tabular}
Use the given table to complete the statements.
The parent function of the function represented in the table is $\square$
If function $f$ was translated down 4 units, the $\square$ $f(x)$ -values would be $\square$
A point in the table for the transformed function would be $\square$ .
Answer (1)
The parent function is identified as 3 x .
Translating the function down by 4 units means subtracting 4 from the f ( x ) values.
The f ( x ) values are decreased by 4.
A point on the transformed function is ( 1 , 9 ) .
Explanation
Analyzing the Problem We are given a table of values for a function f ( x ) and asked to identify the parent function, describe the transformation when f ( x ) is translated down 4 units, and find a point on the transformed function.
Identifying a Pattern First, let's analyze the given values to identify a pattern. The table contains the following values: f ( 1 ) = 13 , f ( 2 ) = 19 , f ( 3 ) = 37 , f ( 4 ) = 91 , f ( 5 ) = 253 .
Calculating Differences Let's calculate the differences between consecutive f ( x ) values: 19 − 13 = 6 , 37 − 19 = 18 , 91 − 37 = 54 , 253 − 91 = 162 . Observe that the differences are multiples of 6: 6 = 6 ⋅ 1 , 18 = 6 ⋅ 3 , 54 = 6 ⋅ 9 , 162 = 6 ⋅ 27 . The factors 1, 3, 9, 27 suggest a power of 3.
Finding the Function Consider the function g ( x ) = 3 x + 10 . Let's check if it matches the given values: g ( 1 ) = 3 1 + 10 = 13 g ( 2 ) = 3 2 + 10 = 9 + 10 = 19 g ( 3 ) = 3 3 + 10 = 27 + 10 = 37 g ( 4 ) = 3 4 + 10 = 81 + 10 = 91 g ( 5 ) = 3 5 + 10 = 243 + 10 = 253 Since g ( x ) matches all the given values, we can conclude that f ( x ) = 3 x + 10 .
Describing the Transformation The parent function is 3 x . If f ( x ) is translated down 4 units, the new function is f ( x ) − 4 . The f ( x ) values would be decreased by 4. A point on the transformed function would be ( 1 , 13 − 4 ) = ( 1 , 9 ) .
Final Answer Therefore, the parent function of the function represented in the table is 3 x . If function f was translated down 4 units, the f ( x ) -values would be decreased by 4. A point in the table for the transformed function would be ( 1 , 9 ) .
Examples
Imagine you're tracking the growth of a plant over several days. The height of the plant each day follows an exponential pattern similar to the function in the table. Understanding the parent function (the basic exponential growth) helps you predict future growth. If you decide to trim the plant by a fixed amount each day (like translating the function down), you can easily calculate the new height of the plant each day.
