Question

Complete the tables to show the parent function y=f(x)
and its translation y=f(x+5)
. (2 points)
Original Function
x
y=f(x)
−2
5
−1
2
0 1
1 2
2 5

Answer 1

To find the translation of the parent function \( y = f(x) \) to \( y = f(x + 5) \), we replace \( x \) with \( x + 5 \). This essentially shifts the graph of the function to the left by 5 units.

Let's work through the provided values for \( y = f(x) \) and calculate the new \( x \) values for \( y = f(x + 5) \).

**Original Function:**
| \( x \) | \( y = f(x) \) |
|:---------:|:-----------------:|
| -2 | 5 |
| -1 | 2 |
| 0 | 1 |
| 1 | 2 |
| 2 | 5 |

**Translated Function:**
To find \( y = f(x + 5) \), we compute the new \( x \) values by subtracting 5 from each original \( x \).

| \( x \) | \( y = f(x + 5) \) |
|:---------:|:---------------------:|
| -7 | 5 | (since \( f(-2) = 5 \))
| -6 | 2 | (since \( f(-1) = 2 \))
| -5 | 1 | (since \( f(0) = 1 \))
| -4 | 2 | (since \( f(1) = 2 \))
| -3 | 5 | (since \( f(2) = 5 \))

So, the completed table is:

**Translation Function:**
| \( x \) | \( y = f(x + 5) \) |
|:---------:|:---------------------:|
| -7 | 5 |
| -6 | 2 |
| -5 | 1 |
| -4 | 2 |
| -3 | 5 |

In summary:
- For every \( x \) value in the original function, subtracting 5 gives the corresponding \( x \) value in the translated function, while the \( y \) values remain the same.