The table provided contains input values (IN $x$) and output values (OUT $y$). The goal is to determine the pattern or rule that connects $x$ to $y$, use it to complete the table, and explain the rule in words.

Provided Data:

Step 1: Identify the Pattern

We compare the known pairs of $x$ and $y$:

• For $x = 2$, $y = 9$
• For $x = 10$, $y = 25$
• For $x = 6$, $y = 17$

The differences in $y$ relative to $x$ suggest a quadratic relationship, i.e., $y = ax^2 + bx + c$. We'll derive the equation step-by-step.

Step 2: Solve for the Rule

Using the known pairs:

1. $y(2) = 9 \Rightarrow 4a + 2b + c = 9$
2. $y(10) = 25 \Rightarrow 100a + 10b + c = 25$
3. $y(6) = 17 \Rightarrow 36a + 6b + c = 17$

Now, solve this system of equations to find $a$, $b$, and $c$.

Subtract Equations to Simplify:

• From (2) and (1): $(100a + 10b + c) - (4a + 2b + c) = 25 - 9$ 96a + 8b = 16 \quad \Rightarrow \quad 12a + b = 2 \tag{4}

• From (3) and (1): $(36a + 6b + c) - (4a + 2b + c) = 17 - 9$ 32a + 4b = 8 \quad \Rightarrow \quad 8a + b = 2 \tag{5}

Solve the Two Linear Equations:

• Subtract (5) from (4): $(12a + b) - (8a + b) = 2 - 2$ $4a = 0 \quad \Rightarrow \quad a = 0$

• Substitute $a = 0$ into (4): $12(0) + b = 2 \quad \Rightarrow \quad b = 2$

• Use $a = 0$ and $b = 2$ in (1): $4(0) + 2(2) + c = 9 \quad \Rightarrow \quad c = 5$

Step 3: Write the Equation

The equation is: $y = 2x + 5$

Step 4: Complete the Table

Use $y = 2x + 5$ to calculate the missing values.

Step 5: Explanation in Words

The rule for the table is:

1. Multiply the input ($x$) by 2.
2. Add 5 to the result.
3. The final result is the output ($y$).

For example:

• If $x = 7$, then $y = 2(7) + 5 = 19$.
• If $x = -3$, then $y = 2(-3) + 5 = -1$.

Do you have any questions or want more details?

Here are 5 related questions to expand your understanding:

1. How can you confirm that a relationship between $x$ and $y$ is linear?
2. What happens to the output ($y$) if the input ($x$) is doubled?
3. How would the table change if the rule was $y = x^2 + 2x + 5$?
4. How do you solve systems of equations with three variables step-by-step?
5. Can you apply this method to find rules for exponential relationships?

Tip: Always check your derived equation by plugging in the known values of $x$ and comparing the outputs!