Question

Complete the table to show the parent function y = f(x) and its translation y = f(x + 5)
Original function
x y = f(x)
-2 5
-1 2
0 1
1 2
2 5

Translated function
x y = (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 how the input values will change based on the translation. The translation \( y = f(x + 5) \) means that we take the original input \( x \) and subtract 5 to find the corresponding output of \( f \).

Let's find the new \( x \) values for each of the original \( y = f(x) \) pairs:

1. For \( y = 5 \) (originally when \( x = -2 \)):
\[
x + 5 = -2 \implies x = -2 - 5 = -7
\]

2. For \( y = 2 \) (originally when \( x = -1 \)):
\[
x + 5 = -1 \implies x = -1 - 5 = -6
\]

3. For \( y = 1 \) (originally when \( x = 0 \)):
\[
x + 5 = 0 \implies x = 0 - 5 = -5
\]

4. For \( y = 2 \) (originally when \( x = 1 \)):
\[
x + 5 = 1 \implies x = 1 - 5 = -4
\]

5. For \( y = 5 \) (originally when \( x = 2 \)):
\[
x + 5 = 2 \implies x = 2 - 5 = -3
\]

Now we can complete the translated function table:

| Translated function | x | y = f(x + 5) |
|---------------------|-----|---------------|
| | -7 | 5 |
| | -6 | 2 |
| | -5 | 1 |
| | -4 | 2 |
| | -3 | 5 |

So the completed table for the translated function \( y = f(x + 5) \) looks like this:

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

| Translated function | x | y = f(x + 5) |
|---------------------|-----|---------------|
| | -7 | 5 |
| | -6 | 2 |
| | -5 | 1 |
| | -4 | 2 |
| | -3 | 5 |