Number Patterns - Grade 5 - Free Printable

Problem: Complete the function tables as per the given rules.

#### Step 1: Solve the first set of function tables

1. Rule: Subtract 4
- Formula: \( Y = X - 4 \)
- Table:
| X | Y |
|---|---|
| 4 | 12 |
| | |
| 10| |

- For \( X = 4 \):
\[
Y = 4 - 4 = 0
\]
So, the entry for \( X = 4 \) is incorrect in the table. It should be \( Y = 0 \).

- For \( X = 10 \):
\[
Y = 10 - 4 = 6
\]

- Updated table:
| X | Y |
|---|---|
| 4 | 0 |
| | |
| 10| 6 |

2. Rule: Add 7; divide by 2
- Formula: \( Y = \frac{X + 7}{2} \)
- Table:
| X | Y |
|---|---|
| | 4 |
| 3 | |
| 5 | |

- For \( Y = 4 \):
\[
4 = \frac{X + 7}{2} \implies 8 = X + 7 \implies X = 1
\]

- For \( X = 3 \):
\[
Y = \frac{3 + 7}{2} = \frac{10}{2} = 5
\]

- For \( X = 5 \):
\[
Y = \frac{5 + 7}{2} = \frac{12}{2} = 6
\]

- Updated table:
| X | Y |
|---|---|
| 1 | 4 |
| 3 | 5 |
| 5 | 6 |

3. Rule: Multiply by 3; add 1
- Formula: \( Y = 3X + 1 \)
- Table:
| X | Y |
|---|---|
| | 1 |
| | 13 |
| 12| |

- For \( Y = 1 \):
\[
1 = 3X + 1 \implies 0 = 3X \implies X = 0
\]

- For \( Y = 13 \):
\[
13 = 3X + 1 \implies 12 = 3X \implies X = 4
\]

- For \( X = 12 \):
\[
Y = 3(12) + 1 = 36 + 1 = 37
\]

- Updated table:
| X | Y |
|---|---|
| 0 | 1 |
| 4 | 13 |
| 12| 37 |

#### Step 2: Solve the second set of function tables using equations

1. Rule: \( x - 5 \)
- Formula: \( Y = X - 5 \)
- Table:
| Input (x) | Output (y) |
|-----------|------------|
| 10 | |
| 12 | |
| 14 | |

- For \( X = 10 \):
\[
Y = 10 - 5 = 5
\]

- For \( X = 12 \):
\[
Y = 12 - 5 = 7
\]

- For \( X = 14 \):
\[
Y = 14 - 5 = 9
\]

- Updated table:
| Input (x) | Output (y) |
|-----------|------------|
| 10 | 5 |
| 12 | 7 |
| 14 | 9 |

2. Rule: \( x + 2 \)
- Formula: \( Y = X + 2 \)
- Table:
| Input (x) | Output (y) |
|-----------|------------|
| 3 | |
| 4 | |
| 5 | |

- For \( X = 3 \):
\[
Y = 3 + 2 = 5
\]

- For \( X = 4 \):
\[
Y = 4 + 2 = 6
\]

- For \( X = 5 \):
\[
Y = 5 + 2 = 7
\]

- Updated table:
| Input (x) | Output (y) |
|-----------|------------|
| 3 | 5 |
| 4 | 6 |
| 5 | 7 |

3. Rule: \( 4x - 3 \)
- Formula: \( Y = 4X - 3 \)
- Table:
| Input (x) | Output (y) |
|-----------|------------|
| 1 | |
| 2 | |
| 3 | |

- For \( X = 1 \):
\[
Y = 4(1) - 3 = 4 - 3 = 1
\]

- For \( X = 2 \):
\[
Y = 4(2) - 3 = 8 - 3 = 5
\]

- For \( X = 3 \):
\[
Y = 4(3) - 3 = 12 - 3 = 9
\]

- Updated table:
| Input (x) | Output (y) |
|-----------|------------|
| 1 | 1 |
| 2 | 5 |
| 3 | 9 |

4. Rule: \( 6x - 10 \)
- Formula: \( Y = 6X - 10 \)
- Table:
| Input (x) | Output (y) |
|-----------|------------|
| 6 | |
| 7 | |
| 8 | |

- For \( X = 6 \):
\[
Y = 6(6) - 10 = 36 - 10 = 26
\]

- For \( X = 7 \):
\[
Y = 6(7) - 10 = 42 - 10 = 32
\]

- For \( X = 8 \):
\[
Y = 6(8) - 10 = 48 - 10 = 38
\]

- Updated table:
| Input (x) | Output (y) |
|-----------|------------|
| 6 | 26 |
| 7 | 32 |
| 8 | 38 |

5. Rule: \( 7x + 9 \)
- Formula: \( Y = 7X + 9 \)
- Table:
| Input (x) | Output (y) |
|-----------|------------|
| 2 | |
| 3 | |
| 4 | |

- For \( X = 2 \):
\[
Y = 7(2) + 9 = 14 + 9 = 23
\]

- For \( X = 3 \):
\[
Y = 7(3) + 9 = 21 + 9 = 30
\]

- For \( X = 4 \):
\[
Y = 7(4) + 9 = 28 + 9 = 37
\]

- Updated table:
| Input (x) | Output (y) |
|-----------|------------|
| 2 | 23 |
| 3 | 30 |
| 4 | 37 |

6. Rule: \( 8x + 2 \)
- Formula: \( Y = 8X + 2 \)
- Table:
| Input (x) | Output (y) |
|-----------|------------|
| 1 | |
| 5 | |
| 10 | |

- For \( X = 1 \):
\[
Y = 8(1) + 2 = 8 + 2 = 10
\]

- For \( X = 5 \):
\[
Y = 8(5) + 2 = 40 + 2 = 42
\]

- For \( X = 10 \):
\[
Y = 8(10) + 2 = 80 + 2 = 82
\]

- Updated table:
| Input (x) | Output (y) |
|-----------|------------|
| 1 | 10 |
| 5 | 42 |
| 10 | 82 |

Final Answer:

\[
\boxed{
\begin{array}{c}
\text{Subtract 4:} \\
\begin{array}{|c|c|}
\hline
X & Y \\
\hline
4 & 0 \\
\hline
10 & 6 \\
\hline
\end{array} \\
\text{Add 7; divide by 2:} \\
\begin{array}{|c|c|}
\hline
X & Y \\
\hline
1 & 4 \\
\hline
3 & 5 \\
\hline
5 & 6 \\
\hline
\end{array} \\
\text{Multiply by 3; add 1:} \\
\begin{array}{|c|c|}
\hline
X & Y \\
\hline
0 & 1 \\
\hline
4 & 13 \\
\hline
12 & 37 \\
\hline
\end{array} \\
\text{x - 5:} \\
\begin{array}{|c|c|}
\hline
Input (x) & Output (y) \\
\hline
10 & 5 \\
\hline
12 & 7 \\
\hline
14 & 9 \\
\hline
\end{array} \\
\text{x + 2:} \\
\begin{array}{|c|c|}
\hline
Input (x) & Output (y) \\
\hline
3 & 5 \\
\hline
4 & 6 \\
\hline
5 & 7 \\
\hline
\end{array} \\
\text{4x - 3:} \\
\begin{array}{|c|c|}
\hline
Input (x) & Output (y) \\
\hline
1 & 1 \\
\hline
2 & 5 \\
\hline
3 & 9 \\
\hline
\end{array} \\
\text{6x - 10:} \\
\begin{array}{|c|c|}
\hline
Input (x) & Output (y) \\
\hline
6 & 26 \\
\hline
7 & 32 \\
\hline
8 & 38 \\
\hline
\end{array} \\
\text{7x + 9:} \\
\begin{array}{|c|c|}
\hline
Input (x) & Output (y) \\
\hline
2 & 23 \\
\hline
3 & 30 \\
\hline
4 & 37 \\
\hline
\end{array} \\
\text{8x + 2:} \\
\begin{array}{|c|c|}
\hline
Input (x) & Output (y) \\
\hline
1 & 10 \\
\hline
5 & 42 \\
\hline
10 & 82 \\
\hline
\end{array}
\end{array}
}
\]