To find the equation of a linear function in slope-intercept form $y = mx + b$, we need to determine two key components:

1. Slope $m$
2. Y-intercept $b$

Step 1: Find the slope $m$

The slope $m$ is calculated using the formula:

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

Using two points from the table, say $(1, -8)$ and $(2, -13)$, we substitute into the formula:

$m = \frac{-13 - (-8)}{2 - 1} = \frac{-13 + 8}{1} = \frac{-5}{1} = -5$

So, the slope $m = -5$.

Step 2: Find the y-intercept $b$

We can now use the slope $m = -5$ and one of the points from the table (say $(1, -8)$) to find the y-intercept $b$ using the slope-intercept form $y = mx + b$.

Substitute $x = 1$, $y = -8$, and $m = -5$ into $y = mx + b$:

$-8 = -5(1) + b$ $-8 = -5 + b$ $b = -8 + 5 = -3$

Final Equation

Thus, the equation of the linear function is:

$y = -5x - 3$

Would you like further details or have any questions about the steps?

Here are 5 related questions to consider:

1. How would the equation change if the slope were positive?
2. How do you determine the slope if the table had different values?
3. What is the graphical interpretation of the y-intercept $b$?
4. Can you find the equation using any other points from the table?
5. How can you verify the correctness of this equation using the table values?

Tip: When dealing with linear equations, the slope $m$ indicates how much $y$ changes for each unit increase in $x$.