Question

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

Answer 1

To complete the table for the function \( y = f(x + 5) \), we will replace \( x \) in the original function \( y = f(x) \) with \( x + 5 \). This means that for each value of \( x \) in the original function, we will find the corresponding \( y \) value for \( x + 5 \).

Let's first list the original function values (based on the provided x-values):

**Original Function Table \( y = f(x) \)**

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

Now, to create the new table for the translated function \( y = f(x + 5) \), we will use values \( x - 5 \) (since \( f(x + 5) \) shifts the graph to the left by 5 units).

Here's the process:

1. Substitute each \( x \) in \( y = f(x) \) with \( x - 5 \) for the new function.
2. Calculate the output for each substituted value.

Here are the corresponding \( x \) values and their translations:

**Translated Function Table \( y = f(x + 5) \)**

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

Now the completed tables are:

**Original Function Table \( y = f(x) \)**

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

**Translated Function Table \( y = f(x + 5) \)**

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