Question

Complete the tables 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 =f(x +5)
5
2
1
2
5

Answer 1

To complete the tables for the parent function \( y = f(x) \) and its translation \( y = f(x + 5) \), we need to understand that the translation \( f(x + 5) \) shifts the graph of the function \( f(x) \) to the left by 5 units. This means that each \( x \) value in the original function will be decreased by 5 in the translated function.

### Original Function
The original function values are provided as follows:

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

### Translated Function
To find the translated function \( y = f(x + 5) \), we will evaluate \( f \) at \( x \) values that are 5 units less than those in the original function table:

1. For \( x = -5 \), \( y = f(-5) = f(-2) = 5 \).
2. For \( x = -4 \), \( y = f(-4) = f(-1) = 2 \).
3. For \( x = -5 \), \( y = f(-5) = f(0) = 1 \).
4. For \( x = -3 \), \( y = f(-3) = f(1) = 2 \).
5. For \( x = -2 \), \( y = f(-2) = f(2) = 5 \).

The completed tables will look like this:

### Translated Function Table

| x | y = f(x + 5) |
|----|--------------|
| -5 | 5 |
| -4 | 2 |
| -3 | 1 |
| -2 | 2 |
| -1 | 5 |

So, the completed function tables are:

### Original Function

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

### Translated Function

| x | y = f(x + 5) |
|----|--------------|
| -5 | 5 |
| -4 | 2 |
| -3 | 1 |
| -2 | 2 |
| -1 | 5 |