To find the equation of the linear function in slope-intercept form ($y = mx + b$) using the given table, we follow these steps:

Step 1: Find the slope ($m$)

The slope ($m$) is the rate of change of $y$ with respect to $x$. Use the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Select two points from the table, such as $(-1, -2)$ and $(2, 10)$:

$m = \frac{10 - (-2)}{2 - (-1)} = \frac{10 + 2}{2 + 1} = \frac{12}{3} = 4$

Step 2: Solve for the y-intercept ($b$)

Using the slope ($m = 4$) and one point, substitute into $y = mx + b$. Use the point $(2, 10)$:

$10 = 4(2) + b$

$10 = 8 + b \implies b = 2$

Step 3: Write the equation

Substitute $m = 4$ and $b = 2$ into $y = mx + b$:

$y = 4x + 2$

Final Answer:

The equation of the linear function is:

$y = 4x + 2$

Let me know if you want further details or have any questions!

Here are 5 related questions to explore:

1. How can you verify the slope using different pairs of points from the table?
2. How does changing one of the $y$-values in the table affect the equation of the line?
3. What would the graph of $y = 4x + 2$ look like?
4. Can you find the $x$-intercept of the line?
5. How can this method be extended to find the equation of a nonlinear function?

Tip: Always double-check your slope calculation using multiple pairs of points to confirm consistency.