Let's analyze the table first to understand the relationship between $x$ and $y$.

Here is the table again:

Step 1: Check if the data follows a linear or exponential pattern.

• Linear Function: A linear function is of the form $y = mx + b$, where the slope $m$ is constant. The difference in $y$-values should be consistent as $x$ changes by a fixed amount.

Let's calculate the differences between consecutive $y$-values:

• $-23 - (-27) = 4$
• $-19 - (-23) = 4$
• $-15 - (-19) = 4$
• $-11 - (-15) = 4$

The differences between consecutive $y$-values are all 4, which suggests a linear relationship.

Step 2: Find the slope $m$ and the y-intercept $b$.

The slope $m$ is the constant rate of change of $y$ with respect to $x$. Since we found that the differences in $y$ are constant, the slope is 4.

So, $m = 4$.

Now, let's find the intercept $b$. We can use any point from the table. Let's choose $x = -9$ and $y = -27$. Plugging these values into the linear equation $y = mx + b$:

$-27 = 4(-9) + b$ $-27 = -36 + b$ $b = -27 + 36$ $b = 9$

Step 3: Write the linear equation.

Thus, the linear equation that models the data is:

$y = 4x + 9$

Summary:

The linear function that fits the data is $y = 4x + 9$.

Would you like a more detailed explanation of any part of the solution?

Here are 5 related questions you might find interesting:

1. How can you identify a linear relationship just by looking at the differences in $y$-values?
2. How do you calculate the slope from a table of data?
3. What is the method for finding the y-intercept when the slope is known?
4. How do you check if a data set fits an exponential function?
5. What would happen if the differences between $y$-values were not constant?

Tip: To easily spot a linear relationship, look at the changes in $y$ as $x$ changes by a constant amount. If the differences are the same, the relationship is linear.