Question

Complete the tables to show the parent function y = f (x) and its translation y = f (x + 5).

(Original function)
(x) -2 -1 0 1 2
(y = f (x) 5 2 1 2 5

(Translated function)
x _ _ _ _ _
y = f (x + 5) 5 2 1 2 5

Answer 1

To complete the table for the translated function \( y = f(x + 5) \), we need to determine the corresponding \( x \) values that result in the same \( y \) values for the original function \( y = f(x) \).

The translation \( y = f(x + 5) \) means we are shifting the graph of \( f(x) \) to the left by 5 units. This means to find the new \( x \) values for the translated function, we need to subtract 5 from each \( x \) value from the original function.

Here’s the completed table:

### Original Function
\[
\begin{array}{c|ccccc}
x & -2 & -1 & 0 & 1 & 2 \\
\hline
y = f(x) & 5 & 2 & 1 & 2 & 5 \\
\end{array}
\]

### Translated Function
To find the \( x \) values for \( y = f(x + 5) \):
- For \( y = 5 \), \( x + 5 = -2 \) gives \( x = -7 \)
- For \( y = 2 \), \( x + 5 = -1 \) gives \( x = -6 \)
- For \( y = 1 \), \( x + 5 = 0 \) gives \( x = -5 \)
- For \( y = 2 \), \( x + 5 = 1 \) gives \( x = -4 \)
- For \( y = 5 \), \( x + 5 = 2 \) gives \( x = -3 \)

So, the completed translated function table is:

\[
\begin{array}{c|ccccc}
x & -7 & -6 & -5 & -4 & -3 \\
\hline
y = f(x + 5) & 5 & 2 & 1 & 2 & 5 \\
\end{array}
\]

### Summary
- The \( x \) values for \( y = f(x + 5) \): -7, -6, -5, -4, -3
- The corresponding \( y \) values remain the same: 5, 2, 1, 2, 5