The table shows the $x$ and $y$ values of a linear function. To find the equation of the line in slope-intercept form $y = mx + b$, we proceed as follows:

Step 1: Find the slope ($m$)

The slope is calculated using the formula: $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

Choose two points from the table, for example, $(1, -1)$ and $(2, 6)$: $m = \frac{6 - (-1)}{2 - 1} = \frac{6 + 1}{1} = \frac{7}{1} = 7$

So, the slope is: $m = 7$

Step 2: Find the $y$-intercept ($b$)

Using the slope $m = 7$ and the point $(0, -8)$ (from the table, where $x = 0$): $y = mx + b$

Substitute $y = -8$, $m = 7$, and $x = 0$: $-8 = 7(0) + b$ $b = -8$

Step 3: Write the equation

Now substitute $m = 7$ and $b = -8$ into $y = mx + b$: $y = 7x - 8$

Final Answer:

$y = 7x - 8$

Would you like further clarifications or practice on related problems? Here are some follow-up questions:

1. How can this equation be verified with other points from the table?
2. What is the significance of the slope in this equation?
3. How can the graph of this function be sketched?
4. What are the x- and y-intercepts of this equation?
5. How do we determine if a table of values represents a linear function?

Tip: To verify the equation, plug other points from the table into $y = 7x - 8$ and check if the $y$-values match!